We can extend our table of sines and cosines of common angles to tangents. You don't have to remember all this information if you can just remember the ratios of the sides of a 45°-45°-90° triangle and a 30°-60°-90° triangle. The ratios are the values of the trig functions.
Note that the tangent of a right angle is listed as infinity. That's because as the angle grows toward 90°, it's tangent grows without bound. It may be better to say that the tangent of 90° is undefined since, using the circle definition, the ray out from the origin at 90° never meets the tangent line.
Thursday, April 23, 2009
Wednesday, April 22, 2009
Angles of elevation and depression!
The term "angle of elevation" refers to the angle above the horizontal from the viewer. If you're at point A, and AH is a horizontal line, then the angle of elevation to a point B above the horizon is the angle BAH. Likewise, the "angle of depression" to a point C below the horizon is the angle CAH.
Tangents are frequently used to solve problems involving angles of elevation and depression.
Common angles
We can extend our table of sines and cosines of common angles to tangents. You don't have to remember all this information if you can just remember the ratios of the sides of a 45°-45°-90° triangle and a 30°-60°-90° triangle. The ratios are the values of the trig functions.
Note that the tangent of a right angle is listed as infinity. That's because as the angle grows toward 90°, it's tangent grows without bound. It may be better to say that the tangent of 90° is undefined since, using the circle definition, the ray out from the origin at 90° never meets the tangent line.
Tangents are frequently used to solve problems involving angles of elevation and depression.
Common angles
We can extend our table of sines and cosines of common angles to tangents. You don't have to remember all this information if you can just remember the ratios of the sides of a 45°-45°-90° triangle and a 30°-60°-90° triangle. The ratios are the values of the trig functions.
Note that the tangent of a right angle is listed as infinity. That's because as the angle grows toward 90°, it's tangent grows without bound. It may be better to say that the tangent of 90° is undefined since, using the circle definition, the ray out from the origin at 90° never meets the tangent line.
Slopes of lines!
One reason that tangents are so important is that they give the slopes of straight lines. Consider the straight line drawn in the x-y coordinate plane.
The point B is where the line cuts the y-axis. We can let the coordinates of B be (0,b) so that b, called the y-intercept, indicates how far above the x-axis B lies. (This notation conflicts with labeling the sides of a triangle a, b, and c, so let's not label the sides right now.)
You can see that the point 1 unit to the right of the origin is labeled 1, and its coordinates, of course, are (1,0). Let C be the point where that verical line cuts the horizontal line through B. Then C has coordinates (1,b).
The point A is where the vertical line above 1 cuts the original line. Let m denote the distance that A is above C. Then A has coordinates (1,b+m). This value m is called the slope of the line. If you move right one unit anywhere along the line, then you'll move up m units.
Now consider the angle CBA. Let's call it the angle of slope. It's tangent is CA/BC = m/1 = m. Therefore, the slope is the tangent of the angle of slope.
The point B is where the line cuts the y-axis. We can let the coordinates of B be (0,b) so that b, called the y-intercept, indicates how far above the x-axis B lies. (This notation conflicts with labeling the sides of a triangle a, b, and c, so let's not label the sides right now.)You can see that the point 1 unit to the right of the origin is labeled 1, and its coordinates, of course, are (1,0). Let C be the point where that verical line cuts the horizontal line through B. Then C has coordinates (1,b).
The point A is where the vertical line above 1 cuts the original line. Let m denote the distance that A is above C. Then A has coordinates (1,b+m). This value m is called the slope of the line. If you move right one unit anywhere along the line, then you'll move up m units.
Now consider the angle CBA. Let's call it the angle of slope. It's tangent is CA/BC = m/1 = m. Therefore, the slope is the tangent of the angle of slope.
Tangents and right triangles!
Just as the sine and cosine can be found as ratios of sides of right triangles, so can the tangent.
We'll use three relations we already have. First, tan A = sin A / cos A. Second, sin A = a/c. Third, cos A = b/c. Dividing a/c by b/c and canceling the c's that appear, we conclude that tan A = a/b. That means that the tangent is the opposite side divided by the adjacent side: tan=opp/adj.
We'll use three relations we already have. First, tan A = sin A / cos A. Second, sin A = a/c. Third, cos A = b/c. Dividing a/c by b/c and canceling the c's that appear, we conclude that tan A = a/b. That means that the tangent is the opposite side divided by the adjacent side: tan=opp/adj.
Tuesday, April 21, 2009
Tangents and slopes!
The definition of the tangent
Sine and cosine are not the only trigonometric functions used in trigonometry. Many others have been used throughout the ages, things like haversines and spreads. The most useful of these is the tangent. In terms of the unit circle diagram, the tangent is the length of the vertical line ED tangent to the circle from the point of tangency E to the point D where that tangent line cuts the ray AD forming the angle.
Drag the point B around to see how the sine, cosine, and tangent change as you change the angle.
Tangent in terms of sine and cosine
Since the two triangles ADE and ABC are similar, we have
ED / AE = CB / AC.
But ED = tan A, AE = 1, CB = sin A, and AC = cos AB. Therefore we have derived the fundamental identity:-
tanA=sinA/cosA.
Sine and cosine are not the only trigonometric functions used in trigonometry. Many others have been used throughout the ages, things like haversines and spreads. The most useful of these is the tangent. In terms of the unit circle diagram, the tangent is the length of the vertical line ED tangent to the circle from the point of tangency E to the point D where that tangent line cuts the ray AD forming the angle.
Drag the point B around to see how the sine, cosine, and tangent change as you change the angle.
Tangent in terms of sine and cosine
Since the two triangles ADE and ABC are similar, we have
ED / AE = CB / AC.
But ED = tan A, AE = 1, CB = sin A, and AC = cos AB. Therefore we have derived the fundamental identity:-
tanA=sinA/cosA.
Definition of cosine
The cosine of an angle is defined as the sine of the complementary angle. The complementary angle equals the given angle subtracted from a right angle, 90°. For instance, if the angle is 30°, then its complement is 60°. Generally, for any angle t,
cos t = sin (90° – t).
Written in terms of radian measurement, this identity becomes
cos t = sin (/2 – t).
Right triangles and cosines
cos t = sin (90° – t).
Written in terms of radian measurement, this identity becomes
cos t = sin (/2 – t).
Right triangles and cosines
Consider a right triangle ABC with a right angle at C. As mentioned before, we'll generally use the letter a to denote the side opposite angle A, the letter b to denote the side opposite angle B, and the letter c to denote the side opposite angle C. Since the sum of the angles in a triangle equals 180°, and angle C is 90°, that means angles A and B add up to 90°, that is, they are complementary angles. Therefore the cosine of B equals the sine of A. We saw on the last page that sin A was the opposite side over the hypotenuse, that is, a/c. Hence, cos B equals a/c. In other words, the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse:
cos=adj/hyp.
Also, cos A = sin B = b/c.
The Pythagorean identity for sines and cosines
Recall the Pythagorean theorem for right triangles. It says that
a2 + b2 = c2
where c is the hypotenuse. This translates very easily into a Pythagorean identity for sines and cosines. Divide both sides by c2 and you get
a2/c2 + b2/c2 = 1.
But a2/c2 = (sin A)2, and b2/c2 = (cos A)2. In order to reduce the number of parentheses that have to be written, it is a convention that the notation sin2 A is an abbreviation for (sin A)2, and similarly for powers of the other trig functions. Thus, we have proven that
sin2 A + cos2 A = 1
when A is an acute angle. We haven't yet seen what sines and cosines of other angles should be, but when we do, we'll have for any angle t one of most important trigonometric identities, the Pythagorean identity for sines and cosines:
sin2 t + cos2 t = 1.
a2 + b2 = c2
where c is the hypotenuse. This translates very easily into a Pythagorean identity for sines and cosines. Divide both sides by c2 and you get
a2/c2 + b2/c2 = 1.
But a2/c2 = (sin A)2, and b2/c2 = (cos A)2. In order to reduce the number of parentheses that have to be written, it is a convention that the notation sin2 A is an abbreviation for (sin A)2, and similarly for powers of the other trig functions. Thus, we have proven that
sin2 A + cos2 A = 1
when A is an acute angle. We haven't yet seen what sines and cosines of other angles should be, but when we do, we'll have for any angle t one of most important trigonometric identities, the Pythagorean identity for sines and cosines:
sin2 t + cos2 t = 1.
The standard notation for a right triangle!
For a while we'll be looking mainly at right triangles, so it would be useful to use a standard notation for the angles and sides of these triangles.
Consider a right triangle ABC with a right angle at C. We'll generally use the letter a to denote the side opposite angle A, the letter b to denote the side opposite angle B, and the letter c to denote the side opposite angle C, that is, the hypotenuse.
With this notation, sin A = a/c, and sin B = b/c.
Next we'll look at cosines. Cosines are just sines of the complementary angle. Thus, the name "cosine" ("co" being the first two letters of "complement"). For triangle ABC above, cos A is just sin B.
We can use properties of similar triangles to relate sines to right triangles. In the figure above the triangle ABC is a right triangle with a right angle at angle C and a hypotenuse of length 1. Consider a similar right triangle AB'C' with a hypotenuse of arbitrary length. (If your web browser is Java-enabled, you can drag the points B' to change the size of the right triangle AB'C'.)
Since the triangles are similar, the ratio BC to AB equals the ratio B'C' to AB'. But AB equals 1. Hence,
BC = B'C' / AB'
but BC = sin A, so
sin A = B'C' / AB'
This result is most easily remembered as the sine of an angle in a right triangle equals the opposite side divided by the hypotenuse: sin= opp/hyp.
BC = B'C' / AB'
but BC = sin A, so
sin A = B'C' / AB'
This result is most easily remembered as the sine of an angle in a right triangle equals the opposite side divided by the hypotenuse: sin= opp/hyp.
The relation between sines and chords
In this section we'll only consider sines of angles between 0° and 90°. In the section on trigonometric functions, we'll define sines for arbitrary angles.
A sine is half of a chord. More accurately, the sine of an angle is half the chord of twice the angle.
A sine is half of a chord. More accurately, the sine of an angle is half the chord of twice the angle.
The bow and arrow diagram
Consider the angle BAD in this figure, and assume that AB is of unit length. Let the point C be the foot of the perpendicular dropped from B to the line AD. Then the sine of angle BAD is defined to be the length of the line BC, and it is written sin BAD. You can double the angle BAD to get the angle BAE, and the chord of angle BAE is BE. Thus, the sine BC of angle BAD is half the chord BE of angle BAE, while the angle BAE is twice the angle BAD. Therefore, as stated before, the sine of an angle is half the chord of twice the angle.
The point of this is just to show that sines are all that difficult to understand. (Whoops, that's a slip! I meant to write "not all that difficult to understand.")
The meaning of the word "sine"The Sanskrit word for chord-half was jya-ardha, which was sometimes shortened to jiva. This was brought into Arabic as jiba, and written in Arabic simply with two consonants jb, vowels not being written. Later, Latin translators selected the word sinus to translate jb thinking that the word was an arabic word jaib, which meant breast, and sinus had breast and bay as two of its meanings. In English, sinus was imported as "sine".
This word history for "sine" is interesting because it follows the path of trigonometry from India, through the Arabic language from Baghdad through Spain, into western Europe in the Latin language, and then to modern languages such as English.
Chords!
What is a chord?
As used in mathematics, the word chord refers to a straight line drawn between two points on a circle (or more generally, on any curve). The known first trigonometric table was a table of chords. In modern times, the sine is used instead (sines and chords are closely related), but, perhaps, chords are more intuitive.
For example, the angle AOB in the diagram shows the curve connecting A to B to be an arc of a circle. The straight line AB is the chord. Of course, the length of the chord depends on the radius of the circle, in fact, it is proportional to the radius of the circle.
Trigonometry began with chords
For example, the angle AOB in the diagram shows the curve connecting A to B to be an arc of a circle. The straight line AB is the chord. Of course, the length of the chord depends on the radius of the circle, in fact, it is proportional to the radius of the circle.
Trigonometry began with chords
Hipparchus (190-120 B.C.E.) produced the first trigonometric table for use in astronomy. It was a table of chords for angles in a circle of large fixed radius. Incidentally, his table was not in terms of degrees, but "steps," each step being 1/24 of a circle. Later, Ptolemy (100-178 C.E.) constructed a more complete table of chords. His table had chords for angles increasing from 1/2 degree to 180 degrees by steps of 1/2 degree. It also included aids for interpolating chords for minutes of angle. Ptolemy used a different large fixed radius than Hipparchus. The advantage of a large radius is that fractions can be avoided. In contrast, our present-day trigonometric functions are based on a unit circle, that is, a circle of radius 1. Of course using a unit circle doesn't avoid fractions, but we have decimal fractions which are easy to work with.
Although trigonometry was, and still could be, based on chords as the primary trigonometric function, a slight modification of chords, called "sines," turns out to be more convenient. Sines were first used in India a few centuries after chords were first used in ancient Greece.
Although trigonometry was, and still could be, based on chords as the primary trigonometric function, a slight modification of chords, called "sines," turns out to be more convenient. Sines were first used in India a few centuries after chords were first used in ancient Greece.
Exercises!
Don't wory, it's not an head-ache!
It's very simple, and to make it even simple, ive given you many hints.
Each set of exercises includes first the statements of the exercises, second some hints to solve the exercises, and third the answer to the exercises.
1. Express the following angles in radians. (a). 12 degrees, 28 minutes, that is, 12° 28'. (b). 36° 12'
[To convert degrees to radians, first convert the number of degrees, minutes, and seconds to decimal form. Divide the number of minutes by 60 and add to the number of degrees. So, for example, 12° 28' is 12 + 28/60 which equals 12.467°. Next multiply by and divide by 180 to get the angle in radians.]
2. Reduce the following numbers of radians to degrees, minutes, and seconds. (a). 0.47623. (b). 0.25412
[Conversely, to convert radians to degrees divide by and multiply by 180. So, 0.47623 divided by and multiplied by 180 gives 27.286°. You can convert the fractions of a degree to minutes and seconds as follows. Multiply the fraction by 60 to get the number of minutes. Here, 0.286 times 60 equals 17.16, so the angle could be written as 27° 17.16'. Then take any fraction of a minute that remains and multiply by 60 again to get the number of seconds. Here, 0.16 times 60 equals about 10, so the angle can also be written as 27° 17' 10".]
3. Given the angle a and the radius r, to find the length of the subtending arc. (a). a = 0° 17' 48", r = 6.2935. (b). a = 121° 6' 18", r = 0.2163.
[In order to find the length of the arc, first convert the angle to radians. For 3(a), 0°17'48" is 0.0051778 radians. Then multiply by the radius to find the length of the arc.]
4. Given the length of the arc l and the radius r, to find the angle subtended at the center. (a). l = .16296, r = 12.587. (b). l = 1.3672, r = 1.2978.
[To find the angle, divide by the radius. That gives you the angle in radians]
5. Given the length of the arc l and the angle a which it subtends at the center, to find the radius. (a). a = 0° 44' 30", l = .032592. (b). a = 60° 21' 6", l = .4572.
[As mentioned above, radian measure times radius = arc length, so, using the letters for this problem, ar = l, but a needs to be converted from degree measurement to radian measurement first. So, to find the radius r, first convert the angle a to radians, then divide that into the length l of the arc.]
6.nd the length to the nearest inch of a circular arc of 11 degrees 48.3 minutes if the radius is 3200 feet.
[Arc length equals radius times the angle in radians.]
7. A railroad curve forms a circular arc of 9 degrees 36.7 minutes, the radius to the center line of the track being 2100 feet. If the gauge is 5 feet, find the difference in length of the two rails to the nearest half-inch.
[It helps to draw the figure. The radius to the outer rail is 2102.5 while the radius to the inner rail is 2097.5.]
8.How much does one change latitude by walking due north one mile, assuming the earth to be a sphere of radius 3956 miles?
[You've got a circle of radius 3956 miles and an arc of that circle of length 1 mile. What is the angle in degrees? (The mean radius of the earth was known fairly accurately in 1914. See if you can find out what Eratosthenes thought the radius of the earth was back in the third century B.C.) ]
9.Compute the length in feet of one minute of arc on a great circle of the earth. How long is the length of one second of arc?
[A minute of arc is 1/60 of a degree. Convert to radians. The radius is 3956. What is the length of the arc? ]
10.On a circle of radius 5.782 meters the length of an arc is 1.742 meters. What angle does it subtend at the center?
[Since the length of the arc equals radius times the angle in radians, it follows that the angle in radians equals the length of the arc divided by the radius. It's easy to convert radians to degrees. ]
11.Imagine that the diameter of the balloon is a part of an arc of a circle with you at the center. (It isn't exactly part of the arc, but it's pretty close.) That arc is 50 feet long. You know the angle, so what is the radius of that circle?
[Imagine that the diameter of the balloon is a part of an arc of a circle with you at the center. (It isn't exactly part of the arc, but it's pretty close.) That arc is 50 feet long. You know the angle, so what is the radius of that circle? ]
Answers
1. (a). 0.2176. (b). 0.6318.
2. (a). 27° 17' 10". (b). 14.56° = 14°33.6' = 14°33'36".
3. (a). 0.03259 (b). 2.1137 times 0.2163 equals 0.4572.
4. (a). 0.16296/12.587 = 0.012947 radians = 0° 44' 30". (b). 1.3672/1.2978 = 1.0535 radians = 60.360° = 60° 21.6' = 60° 21' 35".
5. (a). l/a = .032592/.01294 = 2.518. (b). l/a = .4572/1.0533 = .4340.
6. ra = (3200') (0.20604) = 659.31' = 659' 4".
7. The angle a = 0.16776 radians. The difference in the lengths is 2102.5a – 1997.5a which is 5a. Thus, the answer is 0.84 feet, which to the nearest inch is 10 inches.
9. Angle = 1/3956 = 0.0002528 radians = 0.01448° = 0.8690' = 52.14".
10. One minute = 0.0002909 radians. 1.15075 miles = 6076 feet. Therefore one second will correspond to 101.3 feet.
14. a = l/r = 1.742/5.782 = 0.3013 radians = 17.26° = 17°16'.
23. The angle a is 8.5', which is 0.00247 radians. So the radius is r = l/a = 50/0.00247 = 20222' = 3.83 miles, nearly four miles.
It's very simple, and to make it even simple, ive given you many hints.
Each set of exercises includes first the statements of the exercises, second some hints to solve the exercises, and third the answer to the exercises.
1. Express the following angles in radians. (a). 12 degrees, 28 minutes, that is, 12° 28'. (b). 36° 12'
[To convert degrees to radians, first convert the number of degrees, minutes, and seconds to decimal form. Divide the number of minutes by 60 and add to the number of degrees. So, for example, 12° 28' is 12 + 28/60 which equals 12.467°. Next multiply by and divide by 180 to get the angle in radians.]
2. Reduce the following numbers of radians to degrees, minutes, and seconds. (a). 0.47623. (b). 0.25412
[Conversely, to convert radians to degrees divide by and multiply by 180. So, 0.47623 divided by and multiplied by 180 gives 27.286°. You can convert the fractions of a degree to minutes and seconds as follows. Multiply the fraction by 60 to get the number of minutes. Here, 0.286 times 60 equals 17.16, so the angle could be written as 27° 17.16'. Then take any fraction of a minute that remains and multiply by 60 again to get the number of seconds. Here, 0.16 times 60 equals about 10, so the angle can also be written as 27° 17' 10".]
3. Given the angle a and the radius r, to find the length of the subtending arc. (a). a = 0° 17' 48", r = 6.2935. (b). a = 121° 6' 18", r = 0.2163.
[In order to find the length of the arc, first convert the angle to radians. For 3(a), 0°17'48" is 0.0051778 radians. Then multiply by the radius to find the length of the arc.]
4. Given the length of the arc l and the radius r, to find the angle subtended at the center. (a). l = .16296, r = 12.587. (b). l = 1.3672, r = 1.2978.
[To find the angle, divide by the radius. That gives you the angle in radians]
5. Given the length of the arc l and the angle a which it subtends at the center, to find the radius. (a). a = 0° 44' 30", l = .032592. (b). a = 60° 21' 6", l = .4572.
[As mentioned above, radian measure times radius = arc length, so, using the letters for this problem, ar = l, but a needs to be converted from degree measurement to radian measurement first. So, to find the radius r, first convert the angle a to radians, then divide that into the length l of the arc.]
6.nd the length to the nearest inch of a circular arc of 11 degrees 48.3 minutes if the radius is 3200 feet.
[Arc length equals radius times the angle in radians.]
7. A railroad curve forms a circular arc of 9 degrees 36.7 minutes, the radius to the center line of the track being 2100 feet. If the gauge is 5 feet, find the difference in length of the two rails to the nearest half-inch.
[It helps to draw the figure. The radius to the outer rail is 2102.5 while the radius to the inner rail is 2097.5.]
8.How much does one change latitude by walking due north one mile, assuming the earth to be a sphere of radius 3956 miles?
[You've got a circle of radius 3956 miles and an arc of that circle of length 1 mile. What is the angle in degrees? (The mean radius of the earth was known fairly accurately in 1914. See if you can find out what Eratosthenes thought the radius of the earth was back in the third century B.C.) ]
9.Compute the length in feet of one minute of arc on a great circle of the earth. How long is the length of one second of arc?
[A minute of arc is 1/60 of a degree. Convert to radians. The radius is 3956. What is the length of the arc? ]
10.On a circle of radius 5.782 meters the length of an arc is 1.742 meters. What angle does it subtend at the center?
[Since the length of the arc equals radius times the angle in radians, it follows that the angle in radians equals the length of the arc divided by the radius. It's easy to convert radians to degrees. ]
11.Imagine that the diameter of the balloon is a part of an arc of a circle with you at the center. (It isn't exactly part of the arc, but it's pretty close.) That arc is 50 feet long. You know the angle, so what is the radius of that circle?
[Imagine that the diameter of the balloon is a part of an arc of a circle with you at the center. (It isn't exactly part of the arc, but it's pretty close.) That arc is 50 feet long. You know the angle, so what is the radius of that circle? ]
Answers
1. (a). 0.2176. (b). 0.6318.
2. (a). 27° 17' 10". (b). 14.56° = 14°33.6' = 14°33'36".
3. (a). 0.03259 (b). 2.1137 times 0.2163 equals 0.4572.
4. (a). 0.16296/12.587 = 0.012947 radians = 0° 44' 30". (b). 1.3672/1.2978 = 1.0535 radians = 60.360° = 60° 21.6' = 60° 21' 35".
5. (a). l/a = .032592/.01294 = 2.518. (b). l/a = .4572/1.0533 = .4340.
6. ra = (3200') (0.20604) = 659.31' = 659' 4".
7. The angle a = 0.16776 radians. The difference in the lengths is 2102.5a – 1997.5a which is 5a. Thus, the answer is 0.84 feet, which to the nearest inch is 10 inches.
9. Angle = 1/3956 = 0.0002528 radians = 0.01448° = 0.8690' = 52.14".
10. One minute = 0.0002909 radians. 1.15075 miles = 6076 feet. Therefore one second will correspond to 101.3 feet.
14. a = l/r = 1.742/5.782 = 0.3013 radians = 17.26° = 17°16'.
23. The angle a is 8.5', which is 0.00247 radians. So the radius is r = l/a = 50/0.00247 = 20222' = 3.83 miles, nearly four miles.
Radians and arc length
Radians and arc lengthAn alternate definition of radians is sometimes given as a ratio. Instead of taking the unit circle with center at the vertex of the angle, take any circle with center at the vertex of the angle. Then the radian measure of the angle is the ratio of the length of the subtended arc to the radius of the circle. For instance, if the length of the arc is 3 and the radius of the circle is 2, then the radian measure is 1.5.
The reason that this definition works is that the length of the subtended arc is proportional to the radius of the circle. In particular, the definition in terms of a ratio gives the same figure as that given above using the unit circle. This alternate definition is more useful, however, since you can use it to relate lengths of arcs to angles. The formula for this relation is
radian measure times radius = arc length
For instance, an arc of 0.3 radians in a circle of radius 4 has length 0.3 times 4, that is, 1.2.
The reason that this definition works is that the length of the subtended arc is proportional to the radius of the circle. In particular, the definition in terms of a ratio gives the same figure as that given above using the unit circle. This alternate definition is more useful, however, since you can use it to relate lengths of arcs to angles. The formula for this relation is
radian measure times radius = arc length
For instance, an arc of 0.3 radians in a circle of radius 4 has length 0.3 times 4, that is, 1.2.
Angle measurement!
The concept of angleThe concept of angle is one of the most important concepts in geometry. The concepts of equality, sums, and differences of angles are important and used throughout geometry, but the subject of trigonometry is based on the measurement of angles.
There are two commonly used units of measurement for angles. The more familiar unit of measurement is that of degrees. A circle is divided into 360 equal degrees, so that a right angle is 90°. For the time being, we'll only consider angles between 0° and 360°, but later, in the section on trigonometric functions, we'll consider angles greater than 360° and negative angles.
Degrees may be further divided into minutes and seconds, but that division is not as universal as it used to be. Parts of a degree are now frequently referred to decimally. For instance seven and a half degrees is now usually written 7.5°. Each degree is divided into 60 equal parts called minutes. So seven and a half degrees can be called 7 degrees and 30 minutes, written 7° 30'. Each minute is further divided into 60 equal parts called seconds, and, for instance, 2 degrees 5 minutes 30 seconds is written 2° 5' 30". The division of degrees into minutes and seconds of angle is analogous to the division of hours into minutes and seconds of time.
Usually when a single angle is drawn on a xy-plane for analysis, we'll draw it with the vertex at the origin (0,0), one side of the angle along the x-axis, and the other side above the x-axis.
The other common measurement for angles is radians. For this measurement, consider the unit circle (a circle of radius 1) whose center is the vertex of the angle in question. Then the angle cuts off an arc of the circle, and the length of that arc is the radian measure of the angle. It is easy to convert between degree measurement and radian measurement. The circumference of the entire circle is 2 ( is about 3.14159), so it follows that 360° equals 2 radians. Hence, 1° equals /180 radians, and 1 radian equals 180/ degrees.
There are two commonly used units of measurement for angles. The more familiar unit of measurement is that of degrees. A circle is divided into 360 equal degrees, so that a right angle is 90°. For the time being, we'll only consider angles between 0° and 360°, but later, in the section on trigonometric functions, we'll consider angles greater than 360° and negative angles.
Degrees may be further divided into minutes and seconds, but that division is not as universal as it used to be. Parts of a degree are now frequently referred to decimally. For instance seven and a half degrees is now usually written 7.5°. Each degree is divided into 60 equal parts called minutes. So seven and a half degrees can be called 7 degrees and 30 minutes, written 7° 30'. Each minute is further divided into 60 equal parts called seconds, and, for instance, 2 degrees 5 minutes 30 seconds is written 2° 5' 30". The division of degrees into minutes and seconds of angle is analogous to the division of hours into minutes and seconds of time.
Usually when a single angle is drawn on a xy-plane for analysis, we'll draw it with the vertex at the origin (0,0), one side of the angle along the x-axis, and the other side above the x-axis.
The other common measurement for angles is radians. For this measurement, consider the unit circle (a circle of radius 1) whose center is the vertex of the angle in question. Then the angle cuts off an arc of the circle, and the length of that arc is the radian measure of the angle. It is easy to convert between degree measurement and radian measurement. The circumference of the entire circle is 2 ( is about 3.14159), so it follows that 360° equals 2 radians. Hence, 1° equals /180 radians, and 1 radian equals 180/ degrees.
Similar triangles
Two triangles ABC and DEF are similar if (1) their corresponding angles are equal, that is, angle A equals angle D, angle B equals angle E, and angle C equals angle F, and (2) their sides are proportional, that is, the ratios of the three corresponding sides are equal:AB/DE = BC/EF =CA/FD
An explanation of the Pythagorean theorem!
Proof: Start with the right triangle ABC with right angle at C. Draw a square on the hypotenuse AB, and translate the original triangle ABC along this square to get a congruent triangle A'B'C' so that its hypotenuse A'B' is the other side of the square (but the triangle A'B'C' lies inside the square). Draw perpendiculars A'E and B'F from the points A' and B' down to the line BC. Draw a line AG to complete the square ACEG.
Note that ACEG is a square on the leg AC of the original triangle. Also, the square EFB'C' has side B'C' which is equal to BC, so it equals a square on the leg BC. Thus, what we need to show is that the square ABB'A' is equal to the sum of the squares ACEG and EFB'C'.
But that's pretty easy by cutting and pasting. Start with the big square ABB'A'. Translate the triangle A'B'C' back across the square to triangle ABC, and translate the triangle AA'G across the square to the congruent triangle BB'F. Paste the pieces back together, and you see you've filled up the squares ACEG and EFB'C'. Therefore, ABB'A' = ACEG + EFB'C', as required.
Note that ACEG is a square on the leg AC of the original triangle. Also, the square EFB'C' has side B'C' which is equal to BC, so it equals a square on the leg BC. Thus, what we need to show is that the square ABB'A' is equal to the sum of the squares ACEG and EFB'C'.
But that's pretty easy by cutting and pasting. Start with the big square ABB'A'. Translate the triangle A'B'C' back across the square to triangle ABC, and translate the triangle AA'G across the square to the congruent triangle BB'F. Paste the pieces back together, and you see you've filled up the squares ACEG and EFB'C'. Therefore, ABB'A' = ACEG + EFB'C', as required.
Let's agree again to the standard convention for labeling the parts of a right triangle. Let the right angle be labeled C and the hypotenuse c. Let A and B denote the other two angles, and a and b the sides opposite them, respectively. The Pythagorean theorem is about right triangles, that is, triangles, one of whose angles is a 90° angle. A right triangle is displayed in the diagram to the right. The right angle be labeled C and the hypotenuse c, while A and B denote the other two angles, and a and b the sides opposite them, respectively, often called the legs of a right triangle.
The Pythagorean theorem states that the square of the hypotenuse is the sum of the squares of the other two sides, that is,
c2 = a2 + b2
This theorem is useful to determine one of the three sides of a right triangle if you know the other two. For instance, if two legs are a = 5, and b = 12, then you can determine the hypotenuse c by squaring the lengths of the two legs (25 and 144), adding the two squares together (169), then taking the square root to get the value of c, namely, 13.
Likewise, if you know the hypotenuse and one leg, then you can determine the other. For instance, if the hypotenuse is c = 41, and one leg is a = 9, then you can determine the other leg b as follows. Square the hypotenuse and the first leg (1681 and 81), subtract the square of the first leg from the square of the hypotenuse (1600), then take the square root to get the value of the other leg b, namely 40.
The Pythagorean theorem states that the square of the hypotenuse is the sum of the squares of the other two sides, that is,
c2 = a2 + b2
This theorem is useful to determine one of the three sides of a right triangle if you know the other two. For instance, if two legs are a = 5, and b = 12, then you can determine the hypotenuse c by squaring the lengths of the two legs (25 and 144), adding the two squares together (169), then taking the square root to get the value of c, namely, 13.
Likewise, if you know the hypotenuse and one leg, then you can determine the other. For instance, if the hypotenuse is c = 41, and one leg is a = 9, then you can determine the other leg b as follows. Square the hypotenuse and the first leg (1681 and 81), subtract the square of the first leg from the square of the hypotenuse (1600), then take the square root to get the value of the other leg b, namely 40.
What is trigonometry?
Trigonometry as computational geometry-
Trigonometry began as the computational component of geometry. For instance, one statement of plane geometry states that a triangle is determined by a side and two angles. In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180 degrees (usually written 180°).
Angle measurement and tables-
If there is anything that distinguishes trigonometry from the rest of geometry, it is that trig depends on angle measurement and quantities determined by the measure of an angle. Of course, all of geometry depends on treating angles as quantities, but in the rest of geometry, angles aren't measured, they're just compared or added or subtracted.
Trigonometric functions such as sine, cosine, and tangent are used in computations in trigonometry. These functions relate measurements of angles to measurements of associated straight lines as described later in this short course.
Trig functions are not easy to compute like polynomials are. So much time goes into computing them in ancient times that tables were made for their values. Even with tables, using trig functions takes time because any use of a trig function involves at least one multiplication or division, and, when several digits are involved, even multiplication and division are slow. In the early 17th century computation sped up with the invention of logarithms and soon after slide rules. With the advent of calculators computation has become easy. Tables, logarithms, and slide rules aren't needed in trigonometric computations. All you have to do is enter the numbers and push a few buttons to get the answer. One of the things that used to make learning trig difficult was performing the computations. That's not a problem anymore!
Trigonometry began as the computational component of geometry. For instance, one statement of plane geometry states that a triangle is determined by a side and two angles. In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180 degrees (usually written 180°).
Angle measurement and tables-
If there is anything that distinguishes trigonometry from the rest of geometry, it is that trig depends on angle measurement and quantities determined by the measure of an angle. Of course, all of geometry depends on treating angles as quantities, but in the rest of geometry, angles aren't measured, they're just compared or added or subtracted.
Trigonometric functions such as sine, cosine, and tangent are used in computations in trigonometry. These functions relate measurements of angles to measurements of associated straight lines as described later in this short course.
Trig functions are not easy to compute like polynomials are. So much time goes into computing them in ancient times that tables were made for their values. Even with tables, using trig functions takes time because any use of a trig function involves at least one multiplication or division, and, when several digits are involved, even multiplication and division are slow. In the early 17th century computation sped up with the invention of logarithms and soon after slide rules. With the advent of calculators computation has become easy. Tables, logarithms, and slide rules aren't needed in trigonometric computations. All you have to do is enter the numbers and push a few buttons to get the answer. One of the things that used to make learning trig difficult was performing the computations. That's not a problem anymore!
Applications of Trigonometry!
What can you do with trig? Historically, it was developed for astronomy and geography, but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, trig is used in physics, engineering, and chemistry. Within mathematics, trig is used in primarily in calculus (which is perhaps its greatest application), linear algebra, and statistics. Since these fields are used throughout the natural and social sciences, trig is a very useful subject to know.
1. Astronomy and geography-
Trigonometric tables were created over two thousand years ago for computations in astronomy. The stars were thought to be fixed on a crystal sphere of great size, and that model was perfect for practical purposes. Only the planets moved on the sphere. (At the time there were seven recognized planets: Mercury, Venus, Mars, Jupiter, Saturn, the moon, and the sun. Those are the planets that we name our days of the week after. The earth wasn't yet considered to be a planet since it was the center of the universe, and the outer planets weren't discovered then.) The kind of trigonometry needed to understand positions on a sphere is called spherical trigonometry. Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra. Nonetheless, one application of trigonometry is astronomy.
As the earth is also a sphere, trigonometry is used in geography and in navigation. Ptolemy (100-178) used trigonometry in his Geography and used trigonometric tables in his works. Columbus carried a copy of Regiomontanus' Ephemerides Astronomicae on his trips to the New World and used it to his advantage.
2.Engineering and physics-
Although trigonometry was first applied to spheres, it has had greater application to planes. Surveyors have used trigonometry for centuries. Engineers, both military engineers and otherwise, have used trigonometry nearly as long.Physics lays heavy demands on trigonometry. Optics and statics are two early fields of physics that use trigonometry, but all branches of physics use trigonometry since trigonometry aids in understanding space. Related fields such as physical chemistry naturally use trig.
3.Mathematics and its applications-
Of course, trigonometry is used throughout mathematics, and, since mathematics is applied throughout the natural and social sciences, trigonometry has many applications. Calculus, linear algebra, and statistics, in particular, use trigonometry and have many applications in the all the sciences.
1. Astronomy and geography-
Trigonometric tables were created over two thousand years ago for computations in astronomy. The stars were thought to be fixed on a crystal sphere of great size, and that model was perfect for practical purposes. Only the planets moved on the sphere. (At the time there were seven recognized planets: Mercury, Venus, Mars, Jupiter, Saturn, the moon, and the sun. Those are the planets that we name our days of the week after. The earth wasn't yet considered to be a planet since it was the center of the universe, and the outer planets weren't discovered then.) The kind of trigonometry needed to understand positions on a sphere is called spherical trigonometry. Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra. Nonetheless, one application of trigonometry is astronomy.
As the earth is also a sphere, trigonometry is used in geography and in navigation. Ptolemy (100-178) used trigonometry in his Geography and used trigonometric tables in his works. Columbus carried a copy of Regiomontanus' Ephemerides Astronomicae on his trips to the New World and used it to his advantage.
2.Engineering and physics-
Although trigonometry was first applied to spheres, it has had greater application to planes. Surveyors have used trigonometry for centuries. Engineers, both military engineers and otherwise, have used trigonometry nearly as long.Physics lays heavy demands on trigonometry. Optics and statics are two early fields of physics that use trigonometry, but all branches of physics use trigonometry since trigonometry aids in understanding space. Related fields such as physical chemistry naturally use trig.
3.Mathematics and its applications-
Of course, trigonometry is used throughout mathematics, and, since mathematics is applied throughout the natural and social sciences, trigonometry has many applications. Calculus, linear algebra, and statistics, in particular, use trigonometry and have many applications in the all the sciences.
Monday, April 20, 2009
How to learn Trigonometry!
If you would like to learn a bit about trigonometry, or brush up on it, then read on. These notes are more of an introduction and guide than a full course. For a full course you should take a class or at least read a book.
There are no grades and no tests for you to take, and no transcripts and no awards. There are a few exercises for you to work on. The exercises are the most important aspect of a trigonometry course, or any course in mathematics for that matter.
You should already be familiar with algebra and geometry before learning trigonometry. From algebra, you should be comfortable with manipulating algebraic expressions and solving equations. From geometry, you should know about similar triangles, the Pythagorean theorem, and a few other things, but not a great deal.
Trigonometry is like other mathematics. Take your time. Write things down. Draw figures.
Work out the exercises. There aren't many, so do them all. There are hints if you need them. There are short answers given, too, so you can check to see that you did it right. But remember, the answers are not the goal of doing the exercises. The reason you're doing the exercises is to learn trigonometry. Knowing how to get the answer is your goal.
There are no grades and no tests for you to take, and no transcripts and no awards. There are a few exercises for you to work on. The exercises are the most important aspect of a trigonometry course, or any course in mathematics for that matter.
You should already be familiar with algebra and geometry before learning trigonometry. From algebra, you should be comfortable with manipulating algebraic expressions and solving equations. From geometry, you should know about similar triangles, the Pythagorean theorem, and a few other things, but not a great deal.
Trigonometry is like other mathematics. Take your time. Write things down. Draw figures.
Work out the exercises. There aren't many, so do them all. There are hints if you need them. There are short answers given, too, so you can check to see that you did it right. But remember, the answers are not the goal of doing the exercises. The reason you're doing the exercises is to learn trigonometry. Knowing how to get the answer is your goal.
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