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.
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