Precalculus - Trigonometric Functions: The two historical perspectives of trigonometry incorporate different methods for introducing the trigonometric functions. The first perspective used right triangles, this second perspective is based on the unit circle. Unit Circle Consider the unit circle given by 𝑥 2 + 𝑦 2 = 1 as shown. This unit circle will be used frequently to solve many of our trigonometric problems. It is called the unit circle because there is a distance of 1 from the center to the edge (its radius). We will be combining the unit circle with our knowledge of right triangle trigonometry to help find coordinates (𝑥, 𝑦) that land on the circle. Let’s focus on quadrant I first. (𝑥, 𝑦) 1 𝜃 𝑦 𝑥 For a triangle drawn in quadrant I, follow the radius from the center to the edge and then connect to the 𝑥-axis to finish. For any triangle drawn in quadrant I (or any quadrant) the radius will connect with the circle at some coordinate (𝑥, 𝑦). Let’s see how trigonometry can help us find these coordinates. Since 𝑜 𝑦 𝑎 𝑥 sin 𝜃 = = = 𝑦 and cos 𝜃 = = = 𝑥. ℎ 1 ℎ 1 This means cos 𝜃 = 𝑥 and sin 𝜃 = 𝑦, giving us the statement that (𝑥, 𝑦) = (c With this we can find any coordinate on the edge of the circle by finding sine and cosine of the central angle. We can also use this triangle in quadrant I to find all six trigonometric function. Note: in this section, the variable 𝑡 is used in place of 𝜃. In the definitions of the trigonometric functions, note that the tangent and secant are not defined when 𝑥 = 0. Because 𝑡 = 𝜋 sec = 2 1 0 𝜋 2 corresponds to (𝑥, 𝑦) = (0, 1), it follows that tan 𝜋 2 = 1 0 and are undefined. Similarly, the cotangent and cosecant are not defined when 𝑦 = 0. Because 𝑡 = 0 corresponds to (𝑥, 𝑦) = (1, 0), cot 0 and csc 0 are undefined. If an angle 𝜃 is measured in degrees we shall use the degree symbol. If the angle 𝜃 is measured in radians, then no symbol is used. Since the values of the trig functions are determined by coordinates of the point 𝑃 = (𝑥, 𝑦), then the units used to measure the angle 𝜃 are irrelevant. For example, the point 𝑃 = (0, 1) on the unit circle corresponds to 𝜃 = radians or 𝜃 = 90°. So, sin 𝜋 2 = sin 90° = 1 and cos 𝜋 2 = cos 90° = 0 𝜋 2 Coordinates and Triangles in a Unit Circle Now that we’ve learned how to find the points on the circle using trigonometric functions, we can label the points of the common angle measurements (0°, 30°, 45°, 60°, …) using the values we found in the previous section. In this figure the lines are there to represent each triangle that would be used to find the sine and cosine needed to complete the coordinate. Notice that the lines are always connected to the 𝑥 -axis to make the triangles, never the 𝑦-axis. Notice that as we move around the circle from 0° to 360° the signs in the coordinates change. The signs of the coordinates depends on the quadrant the point is in. This will also affect the sides of the triangles in the circle. For example, a triangle in quadrant III will have both legs as negative numbers. To be able to build these triangles outside quadrant I, we will need to make use of reference angles. The reference angles for Degrees: 𝜃 ′ = 180° − 𝜃 Radians: 𝜃 ′ = 𝜋 − 𝜃 in Quadrants II, III, and IV are shown below. 𝜃 ′ = 𝜃 − 180° 𝜃′ = 𝜃 − 𝜋 To see how a reference angle is used to evaluate a trigonometric function, consider the , as shown in point (𝑥, 𝑦) on the terminal side of figure below. 𝜃 ′ = 360° − 𝜃 𝜃 ′ = 2𝜋 − 𝜃 Outside the Unit Circle Most of the time we will be working with triangles that are unable to fit within a unit circle (they are either too big or too small). If this is the case we can adjust the definitions from earlier by replacing the 1’s (for the radius of the unit circle) with 𝑟 to represent the radius of our not-unit circle. The values of the trigonometric functions of 𝜃 become 𝐬𝐢𝐧 𝜽 = 𝐜𝐬𝐜 𝜽 = 𝒚 𝒓 𝒓 𝒚 = = 𝐨𝐩𝐩 𝐡𝐲𝐩 𝐡𝐲𝐩 𝐨𝐩𝐩 𝐜𝐨𝐬 𝜽 = 𝐬𝐞𝐜 𝜽 = 𝒙 𝒓 𝒓 𝒙 = = 𝐚𝐝𝐣 𝐡𝐲𝐩 𝐡𝐲𝐩 𝐚𝐝𝐣 𝐭𝐚𝐧 𝜽 = 𝐜𝐨𝐭 𝜽 = 𝒚 𝒙 𝒙 𝒚 = = 𝐨𝐩𝐩 𝐚𝐝𝐣 𝐚𝐝𝐣 𝐨𝐩𝐩 Ex 1: Determine the exact values of the six trigonometric function of the real number 𝑡. (Remember (𝑥, 𝑦) = (cos 𝑡 , sin 𝑡) = (cos 𝜃 , sin 𝜃)) Find the point (𝑥, 𝑦) on the unit circle that corresponds to the real number 𝑡. 5𝜋 3 Ex 2: t= Ex 3: t =𝜋 Evaluate (if possible) the sine, cosine, and tangent at the real number. 𝜋 Ex 4: t= 3 Ex 5: (Example 1, pg 272) Evaluate, if possible, the six trigonometric functions of the real number. 𝜋 a) 𝑡 = 6 b) 𝑡 = 5𝜋 4 c) 𝑡 = 𝜋 d) 𝑡 = − 𝜋 3 Domain and Period of Sine and Cosine The Domain of the sine and cosine functions is the set of all real numbers. The Range of sine and cosine are as follows: −1 ≤ sin 𝜃 ≤ 1 and −1 ≤ cos 𝜃 ≤ 1 Because the values of sin(𝑡 + 2𝜋) and cos(𝑡 + 2𝜋) correspond to those of sin 𝑡 and cos 𝑡, sine and cosine are considered periodic functions. Ex 6: (Example 2, pg 274) Evaluate the trigonometric function using its period as an aid. a) sin 13𝜋 6 b) cos − 7𝜋 2 A function 𝑓 is even when 𝑓(– 𝑡) = 𝑓(𝑡), and is odd when 𝑓(– 𝑡) = – 𝑓(𝑡). Use the value of the trigonometric function to evaluate the indicated functions. 4 Ex 7: sin t = Ex 8: cos t = − a) sin (−𝑡) 5 3 4 b) csc (−𝑡) a) cos (−𝑡) b) sec (−𝑡)