Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions. cosθ a. = cosθ a.

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1 Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions A unit circle is a circle with radius = 1 whose center is at the origin. Since we know that the formula for the circumference of a circle is C = 2r, then for the unit circle, the circumference would be C = 2(1) = 2. Thus, one complete revolution around the unit circle moves 2 units. FIND THE EXACT VALUES OF THE TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE Let the point P = (a, b) lie on the unit circle, and let represent the angle created between the positive x-axis (the initial side) and the point P. Then, P = (a, b) = (cos, sin ). In other words, the x-coordinate of any point on the unit circle is equal to cosine of (cos = a), and the y-coordinate of any point on the unit circle is equal to the sine of (sin = b). Then, using what we know from the Quotient Identities, tan = sinθ = b cosθ a Also, from the Reciprocal Identities, csc = The box below summarizes these results: 1 1 = sinθ b and sec = 1 1 = cosθ a., and cot = cosθ a = sinθ b. KNOW THE DOMAIN AND RANGE OF THE TRIGONOMETRIC FUNCTIONS The domain of a function is the set of all real number inputs that give real number outputs. For polynomial functions, the domain is, and for rational functions, the domain is all real numbers except. Example: Find the domain for each trig function of. (In other words, find all the values of that cause the function to be undefined. Then exclude those values from the domain.) a) sin b) cos c) tan d) csc e) sec f) cot

2 Now let's examine the ranges of the trig functions, using the figure on the right. Recall that cos is represented by the x-values as we move around the unit circle. The left-most (and thus smallest) x-value is negative one (-1), and the right-most (and thus largest) x-value is positive one (1). Thus, the range for cos is [-1, 1] (or -1 cos 1). Similarly, recall that sin is represented by the y-values as we move around the unit circle. The bottom-most (smallest) y- value is -1 and the top-most (largest) y-value is 1. So the range for sin is also [-1, 1] (or -1 sin 1). The rest of the ranges are not quite as obvious to find, so I will list them below. The values of csc are either less than (or equal to) -1 or greater than (or equal to) 1. The values of sec are either less than (or equal to) -1 or greater than (or equal to) 1. The range of values that both tan and cot can take is All Real Numbers. The table below summarizes the Domains and Ranges for all six trigonometric functions.

3 USE THE PERIODIC PROPERTIES TO FIND THE EXACT VALUES OF THE TRIGONOMETRIC FUNCTIONS Look at the figure on the left. You can see that if we start on the positive x- 1 3 axis and move to the angle =, we would be at point P, If we did one complete revolution (2 radians) from there, we would have 7 gone 1 = + 2 = radians, but we would still end up at point P. 3 3 Even if we did 5 complete revolutions (10 radians) from there, we would 31 have gone 2 = + 10 = radians, but we would still end up at point P. 3 3 In general, for any multiple of 2 that we add to (or subtract from), we will always end up at the same point on the unit circle. And since the point P represents (cos, sin ), that means that the cosine and sine values are unchanged by moving these complete revolutions around the circle. Thus: Functions of this type are called periodic functions. A function f is called periodic if there is a positive number p such that f( + p) = f() whenever and +p are both in the domain of f. The smallest value that p can take is called the fundamental period of f. For sine and cosine (and thus also their reciprocal functions, and, respectively), the fundamental period is. Tangent and cotangent have a fundamental period (also just called a "period") of. Therefore, tan ( + k) = tan and cot ( + k) = cot. Example: Find the exact value of each expression using the fact that the trig functions are periodic. a) cos 420 b) sin 390 c) csc 9 2 d) cot 17 4

4 USE EVEN-ODD PROPERTIES TO FIND THE EXACT VALUES OF THE TRIGONOMETRIC FUNCTIONS You learned in college algebra (and again in precalculus) that an even function is one such that f(-x) = f(x), and that an odd function is one such that f(-x) = - f(x). Reference the figure to the left. Recall that for the point P = (a, b), a = cos and b = sin. Now look at its mirror image point Q = (a, -b) (this is just the point P flipped down across the x-axis). So the angle between the positive x- axis and the point Q is equal to -. Using these points, we can see that: cos(-) =, which is the as cos(), but sin(-) =, which is of sin(). Therefore, cos(-) = cos(), so cosine is an function, but sin(-) =, so sine is an function. It turns out that cosine (and its reciprocal function, ) are the only functions. All of the remaining functions (,,, and ) are. Example: Use the even-odd properties to find the exact value of each expression. a) cos (-30) b) sin(-135) c) sec (-270) d) sin 3 e) sec (-) f) cot 4

5 THE UNIT CIRCLE The unit circle (shown in the figure to the left) is a very valuable tool. You can see at a glance the cosine and sine values for the four quadrantal angles as well as the three main angles within each quadrant (remember, cos = x-value and sin = y-value). For example, looking at the unit circle, you can quickly tell me the following values: a) cos 7 6 = b) sin 2 3 = 11 c) sin 6 = d) cos 5 4 = You can also quickly identify the values of secant and cosecant, since these are just the of the cosine and sine values, respectively. For example, looking at the unit circle, you can quickly tell me the following values: e) csc 5 6 f) sec 4 3 g) sec 7 4 h) csc 5 3 (rationalize the denominator!) y And with a little bit of calculation effort, we can also relatively easily find the values of tangent x and cotangent x y : i) tan 3 j) cot 4 6

6 As nice as the unit circle is to quickly find the trig values of these angles, the reality is that you will not be given the unit circle for your test! (Yeah, I know, I'm SO MEAN!) But using everything we now know about the trig functions, reference angles, periodicity, and even-odd properties, you can really get away with just knowing the coordinates of the quadrantal angles, plus those of the three angles we have studied in Quadrant I (30, 45, and 60). It is imperative that you learn the following values: Quadrantal Angles = 0 = 360 = 0 radians = 2 radians: corresponds to the point (1, 0) on the unit circle. Thus, cos =, sin =. = 90 = radians: corresponds to the point (0, 1) on the unit circle. Thus, cos =, sin =. 2 = 180 = radians: corresponds to the point (-1, 0) on the unit circle. Thus, cos =, sin =. = 270 = 3 radians: corresponds to the point (0, -1) on the unit circle. Thus, cos =, sin =. 2 Quadrant I Angles = 30 = radians: cos =, sin =. = 45 = radians: cos =, sin =. = 60 = radians: cos =, sin =. 30 = 45 = 60 = sin cos Example: Using only the information on this page, find the exact value of each trigonometric function. a) sin (-45) b) cos (-30) c) sec 2 3 d) csc 4 3 e) cos 7 4 f) tan 11 6