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Unformatted text preview: Unit 9 Calculus of Trigonometric Functions Review of Trigonometric Functions You will have learned in previous courses about the six trigonometric functions - functions which associate numeric values with angles. These are the sine, cosine and tangent functions, as well as the cosecant, secant and cotangent functions. If t is any angle, we denote these functions by sin t , cos t , tan t , csc t , sec t and cot t , respectively. You are probably most familiar with angles being measured in degrees. However, you have most likely also encountered the radian , which is another unit of measurement for angles. In calculus, angles are always measured in radians. The technical definition of the radian measure of an angle is the ratio of the arc length produced by that angle to the radius. Definition 9.1. Consider a segment of a circle. Let r be the radius of the circle and s be the arc length of the segment. Then the angle between the radii producing the segment is t radians, where t = s r . For instance, if we consider a full circle, we know that the arc length (i.e. the perimeter of the circle) is s = 2 πr . So we see that there are t = s r = 2 πr r = 2 π radians in a full circle. Similarly, if we consider a right-angle , we are dealing with a circle segment which is one quarter of a full circle. So the arc length is s = 1 4 (2 πr ) = π 2 r and we see that a right-angle is π 2 radians . Note: If you need to convert from degrees to radians, determine what proportion of a full circle (360 ◦ ) the degree measurement represents, and take that same proportion of 2 π , just as we did above. For instance, for 60 ◦ , we have 60 ◦ = 1 6 (360 ◦ ) = 1 6 (2 π ) radians = π 3 radians . Similarly, if you need 1 to convert from radians to degrees, find the proportion of 2 π represented by the radian measure and take that same proportion of 360 ◦ . For instance, π 4 radians = 1 8 (2 π radians ) = 1 8 (360 ◦ ) = 45 ◦ . Recall that for an angle smaller than a right-angle, the trigonometric function values can be found by considering a right-angled triangle containing that angle. Consider an angle of t radians, with 0 < t < π 2 . Construct a right- angled triangle containing this angle, as shown. Let the side lengths be: r for the hypotenuse, x for the side adjacent to angle t and y for the side opposite angle t . a26 a26 a26 a26 a26 a26 a26 t x y r Then the sine value of the angle is given by the ratio of the length of the opposite side to the length of the hypotenuse, i.e., sin t = opp hyp = y r . And the cosine value of the angle is given by the ratio of the length of the adjacent side to the length of the hypotenuse, i.e. cos t = adj hyp = x r . The other trigonometric functions can all be expressed in terms of the sine and cosine functions. We have: tan t = sin t cos t csc t = 1 sin t sec t = 1 cos t cot t = cos t sin t In this course, since we don’t use calculators, it is necessary to know the trig function values of a few basic angles. First of all, remember that both sinetrig function values of a few basic angles....
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