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4.2 Trigonometric Functions: The Unit Circle Objectives Identify a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle. Use domain and period to evaluate sine and cosine functions, and use a calculator to evaluate trigonometric functions. 2 The Unit Circle 3 The Unit Circle The two historical perspectives of trigonometry incorporate different methods for introducing the trigonometric functions. One such perspective follows and is based on the unit circle. Consider the unit circle given by x2 + y2 = 1 Unit circle as shown at the right. 4 The Unit Circle Imagine wrapping the real number line around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown below. 5 The Unit Circle • As the real number line wraps around the unit circle, each real number t corresponds to a point (x, y) on the circle. • For example, the real number 0 corresponds to the point (1, 0). • Moreover, because the unit circle has a circumference of 2, the real number 2 also corresponds to the point (1, 0). 6 The Unit Circle • In general, each real number t also corresponds to a central angle (in standard position) whose radian measure is t. • With this interpretation of t, the arc length formula s = r (with r = 1) indicates that the real number t is the (directional) length of the arc intercepted by the angle , given in radians. 7 The Trigonometric Functions 8 The Unit Circle Imagine a circle on the coordinate plane, with its center at the origin, and a radius of 1. Choose a point on the circle somewhere in quadrant I. 9 The Unit Circle Connect the origin to the point, and from that point drop a perpendicular to the x-axis. This creates a right triangle with hypotenuse of 1. 10 The Unit Circle The length of its legs are the x- and ycoordinates of the chosen point. Applying the definitions of the trigonometric ratios to this triangle gives y sin( ) y 1 is the angle of rotation 1 y x 11 The Unit Circle The coordinates of the chosen point are the cosine and sine of the angle . This provides a way to define functions sin() and cos() for all real numbers . y sin( ) y 1 (x, y) 1 y x The other trigonometric functions can be defined from these. 12 Trigonometric Functions sin( ) y is the angle of rotation (x, y) 1 y x 13 The Trigonometric Functions From the preceding discussion, it follows that the coordinates x and y are two functions of the real variable t. You can use these coordinates to define the six trigonometric functions of t. sine cosecant cosine secant tangent cotangent These six functions are normally abbreviated sin, csc, cos, sec, tan, and cot, respectively. 14 The Trigonometric Functions 15 Around the Circle As that point moves around the unit circle into quadrants II, III, and IV, the new definitions of the trigonometric functions still hold. 16 Reference Angles • The angles whose terminal sides fall in quadrants II, III, and IV will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in quadrant I. • The acute angle which produces the same values is called the reference angle. 17 Reference Angles • The reference angle is the angle between the terminal side and the nearest arm of the x-axis. • The reference angle is the angle, with vertex at the origin, in the right triangle created by dropping a perpendicular from the point on the unit circle to the x-axis. 18 Quadrant II Original angle For an angle, , in quadrant II, the reference angle is . In quadrant II, Reference angle sin() is positive cos() is negative tan() is negative 19 Quadrant III For an angle, , in quadrant III, the reference angle is -. In quadrant III, Original angle sin() is negative cos() is negative Reference angle tan() is positive 20 Quadrant IV Reference angle For an angle, , in quadrant IV, the reference angle is 2. In quadrant IV, sin() is negative cos() is positive tan() is negative Original angle 21 All Star Trig Class Use the phrase “All Star Trig Class” to remember the signs of the trig functions in different quadrants. Star Sine is positive Trig Tan is positive All All functions are positive Class Cos is positive 22 Unit Circle sin One of the most useful tools in trigonometry is the unit circle. It is a circle, with radius 1 unit, that is on the x-y coordinate plane. 1 cos The x-axis corresponds to the cosine function, and the y-axis corresponds to the sine function. The angles are measured from the positive x-axis (standard position) counterclockwise. In order to create the unit circle, we must use the special right triangles below: 1 45º 1 60º 30º 30º -60º -90º The hypotenuse for each triangle is 1 unit. 45º 45º -45º -90º 23 Unit Circle You first need to find the lengths of the other sides of each right triangle... 1 60º 1 2 30º 3 2 45º 1 2 2 45º 2 2 24 Unit Circle Usefulness of Knowing Trigonometric Functions of Special Angles: 30o, 45o, 60o The trigonometric function values derived from knowing the side ratios of the 30-60-90 and 45-45-90 triangles are “exact” numbers, not decimal approximations as could be obtained from using a calculator You will often be asked to find exact trig function values for angles other than 30o, 45o and 60o angles that are somehow related to trig function values of these angles 25 Now, use the corresponding triangle to find the coordinates on the unit circle... (0, 1) sin What are the coordinates of This cooresponds point? to (costhis 30,sin 30) 3 1 , (Use 2 2 your (cos 30, sin 30) 30-60-90 (–1, 0) 30º 3 2 1 2 triangle) cos (1, 0) (0, –1) Now, use the corresponding triangle to find the coordinates on the unit circle... (0, 1) sin What are the (Use your coordinates of 245-45-90 this point? , 2 (cos45, sin 45) triangle) 2 2 3 1 , 2 2 2 2 (–1, 0) 45º 2 2 (cos 30, sin 30) cos (1, 0) (0, –1) Now, use the corresponding triangle to find the coordinates on the unit circle... What are the of (0, coordinates 1) sin this point? 2 2 , (cos45, 2 2 3 (–1, 0) (Use your 130-60-90 3 triangle) (cos60, sin 60) , 2 2 60 ̊ 1/2 sin 45) 3 1 , 2 2 2 (cos 30, sin 30) cos (1, 0) (0, –1) Use this same technique to complete the unit circle. (0, 1) sin 1 3 , 2 2 (cos60, sin 60) 2 2 , (cos45, 2 2 sin 45) 3 1 , 2 2 (cos 30, sin 30) (–1, 0) cos (1, 0) (0, –1) Unit Circle π 2 (0, 1) (-1, 0) π 0 (1, 0) 3π (0, -1) 2 30 Unit Circle (0, 1) 3 1 ( , ) 2 2 5π π 3 1 ( , ) 6 2 2 6 30° 30° (-1, 0) (1, 0) 30° 30° 7π 6 3 1 ( , ) 2 2 11π 3 1 ( , ) 2 2 6 (0, -1) 31 Unit Circle (0, 1) 2 2 ( , ) 2 2 3π π 2 2 ( , ) 4 2 2 4 45° 45° (-1, 0) (1, 0) 45° 45° 2 2 ( ,) 2 2 5π 4 (0, -1) 7π 2 2 ( ,) 2 2 32 4 Unit Circle 60° (-1, 0) 60° (1, 0) 60° 60° 1 3 ( , ) 2 2 4π 3 π 1 3 ( , ) 3 2 2 (0, 1) 1 3 2π ( , ) 2 2 3 (0, -1) 1 3 5π ( , - ) 2 2 3 33 Unit Circle 34 Function Values of Special Angles Are used a lot in Trigonomet ry and can quickly be determined from memorized triangles . sin cos tan cot sec csc 30 1 2 3 2 45 2 2 2 2 60 3 2 1 2 3 3 3 1 2 3 3 1 3 2 3 3 2 2 2 2 3 3 This chart can also be quickly completed by memorizing the first column and using identities . 35 36 The Trigonometric Functions A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be x y 1 2 2 (0,1) (-1,0) (1,0) (0,-1) So points on this circle must satisfy this equation. 37 The Trigonometric Functions Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value? x= 2 2 x y 1 1/2 You can see there (0,1) 1 3 2 , are two y values. 1 2 2 2 They can be found y 1 by putting 1/2 into 2 (-1,0) (1,0) the equation for x 3 and solving for y. 2 y 4 3 y 2 1 3 , 2 2 (0,-1) We'll look at a larger version of this and make a right triangle. 38 The Trigonometric Functions We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value (0,1) and the hypotenuse is always 1 because it is a radius of the circle. 1 (-1,0) 1 3 , 2 2 3 2 sin (1,0) 1 2 tan (0,-1) cos 3 2 3 1 2 1 21 1 2 3 2 3 1 2 Notice the sine is just the y value of the unit circle point and the cosine is just the x value. 39 So if I want a trig function for whose terminal side contains a point on the unit circle, the y value is the sine, the x value is the cosine and y/x is the tangent. 2 2 , 2 2 (-1,0) (0,1) 1 3 , 2 2 sin (1,0) tan (0,-1) 1 3 , 2 2 cos 2 2 2 2 2 2 1 2 2 We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions. Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division? These are 0,1 easy to 2 2 2 2 90° memorize , 2 2 2 , 2 135° since they 45° all have the 2 sin 225 2 same value with 180° 45° 1,0 0° 1,0 different signs depending 225° on the 2 2 315° 2 2 , 2 , 2 quadrant. 270° 2 2 0,1 We can label this all the way around with how many degrees an angle would be and the point on the unit circle that corresponds with the terminal side of the angle. We could then find any of the trig functions. Can you figure out what these angles would be in radians? 0,1 2 2 2 , 2 1,0 7 sin 4 2 2 135° 3 180° 2 45° 4 4 5 4225° 3 2 2 2 2 , 2 2 2 2 , 2 90° 270° 0° 1,0 7 4 315° 0,1 2 2 , 2 2 The circle is 2 all the way around so half way is . The upper half is divided into 4 pieces so each piece is /4. Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division? You'll need 1 3 0,1 1 3 , to know , 2 2 cos 330 23 2 2 90° these too 120° 60° 3 1 but you , 3 1 2 2 150° , can see 2 2 30° the pattern. 180° 30° 1,0 0° 1,0 210° 3 1 330° 3 , 1 , 2 2 2 2 240° 270° 300° sin 240 3 2 1 3 , 2 2 0,1 1 3 , 2 2 We can again label the points on the circle and the sine is the y value, the cosine is the x value and the tangent is y over x. Can you figure out what the angles would be in radians? 1 3 , 2 2 0,1 1 3 , 2 2 120° 90° 60° 3 1 2 , 3 1 2 2 2 150° , 5 3 2 2 3 30° 6 180° 30° 6 1,0 7 0° 1,0 11 6 210°4 6 5 3 1 3 3 330° 3 , 1 , 2 2 2 2 240° 2 3 270° 300° 1 3 , 2 2 0,1 1 3 , 2 2 It is still halfway around the circle and the upper half is divided into 6 pieces so each piece is /6. The Trigonometric Functions In the definitions of the trigonometric functions, note that the tangent and secant are not defined when x = 0. For instance, because t = /2 corresponds to (x, y) = (0, 1), 1 sec 𝜃 = 𝑥 𝜋 1 sec = = 𝑢𝑛𝑑𝑒𝑓. 2 0 𝑦 tan 𝜃 = 𝑥 𝜋 1 tan = = 𝑢𝑛𝑑𝑒𝑓. 2 0 (0,1) 𝜋 2 (-1,0) (1,0) it follows that tan( /2) and sec( /2) are undefined. Similarly, the cotangent and cosecant are not defined when y = 0. For instance, because t = 0 corresponds to (x, y) = (1, 0), cot 0 and csc 0 are undefined. (0,-1) 45 You should know this. This is a great reference because you can figure out the trig functions of all these angles quickly. 1 3 , 2 2 The Trigonometric Functions Let t be a real number and let (x, y) be the point on the unit circle corresponding to t. The 6 trigonometric functions are: sine (sin); cosine (cos); tangent (tan); cosecant (csc); secant (sec); cotangent (cot) sin t = y csc t = 1/y (y ≠ 0) (undef. at 0 & 𝜋) cos t =x sec t = 1/x (x ≠ 0)(undef. at tan t = y/x (x ≠ 0)(undef. at 𝜋 2 & 𝜋 2 & 3𝜋 ) 2 cot = x/y 3𝜋 ) 2 (y ≠ 0) (undef. at 0 & 𝜋) 47 Example: Evaluating Circular Functions Solution 3π 3π 3π Evaluate sin and tan . , cos 2 2 2 3π An angle of 2 radians intersects the unit circle at the point (0, -1) 3𝜋 sin 𝜃 = 𝑦 so sin = −1 2 3𝜋 cos 𝜃 = 𝑥 so cos =0 2 𝑦 3𝜋 tan 𝜃 = so tan = 𝑢𝑛𝑑𝑒𝑓. 𝑥 2 48 Evaluating Circular Functions Example (a) Find the exact values of (b) Find the exact value of 7π cos 4 and 5π tan 3 7π sin 4 . . 49 Evaluating Circular Functions Solution cos (a) From the figure 7π 2 7π 2 , sin . 4 2 4 2 (b) The angle -5𝜋/3 radians is coterminal with an angle of 𝜋/3 radians. From the figure 3 5π π 2 tan tan 3 3 3 1 2 1 3 , 2 2 50 YOUR TURN: Find the point (x, y) on the unit circle that corresponds to the real number t 3 5 4 5 3 2 1 3 , 2 2 2 2 − , 2 2 1 3 ,− 2 2 (1, 0) 51 Your Turn – Evaluating Trigonometric Functions Evaluate the six trigonometric functions at each real number. a. b. c. t = d. Solution: For each t-value, begin by finding the corresponding point (x, y) on the unit circle. Then use the definitions of trigonometric functions. a. corresponds to the point (x, y) = . 52 Solution cont’d 53 Solution cont’d 54 Solution b. corresponds to the point (x, y) = cont’d . 55 Solution cont’d 56 Solution cont’d c. t = corresponds to the point (x, y) = (–1, 0). sin = y = 0 cos = x = –1 tan = = =0 57 Solution cont’d (–1, 0) is undefined. is undefined. 58 Solution cont’d d. Moving clockwise around the unit circle, it follows that corresponds to the point (x, y) = 59 Solution cont’d 60 YOUR TURN: EVALUATE THE SIX TRIGONOMETRIC FUNCTIONS AT EACH REAL NUMBER, T 7 4 3 4 3 61 Solution: 7𝜋 4 7𝜋 2 sin =− 4 2 7𝜋 2 cos = 4 2 7𝜋 tan = −1 4 7𝜋 csc =− 2 4 7𝜋 sec = 2 4 7𝜋 cot = −1 4 62 Solution: 𝜋 − 3 𝜋 3 sin − = − 3 2 𝜋 2 3 csc − = − 3 3 𝜋 1 cos − = 3 2 𝜋 sec − = 2 3 𝜋 tan − = − 3 3 𝜋 3 cot − = − 3 3 63 Solution: 4𝜋 − 3 4𝜋 3 sin − = 3 2 4𝜋 2 3 csc − = 3 3 4𝜋 1 cos − =− 3 2 4𝜋 sec − = −2 3 4𝜋 tan − =− 3 3 4𝜋 3 cot − =− 3 3 64 Domain and Period of Sine and Cosine 65 Domain of Sine and Cosine Let’s think about the function f() = sin What is the domain? (remember domain means the “legal” things you can put in for ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1 sin 1 (0, 1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for sine? (sine is the y value so what is the lowest and highest y value?) (1, 0) (-1, 0) (0, -1) 66 Domain of Sine and Cosine Let’s think about the function f() = cos What is the domain? (remember domain means the “legal” things you can put in for ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1 cos 1 (0, 1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for cosine? (cosine is the x value so what is the lowest and highest x value?) (-1, 0) (1, 0) (0, -1) 67 Domain of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure 1.19. sin = 𝑦 cos = 𝑥 Figure 1.19 68 Domain of Sine and Cosine Range: Sin and Cos By definition, sin t = y and cos t = x. Because (x, y) is on the unit circle, you know that –1 y 1 and –1 x 1. So, the values of sine and cosine also range between –1 and 1. –1 y 1 –1 sin t 1 –1 and x 1 –1 cos t 1 Range: Sine and Cosine Functions 69 Period of Sine and Cosine Adding 2 to each value of t in the interval [0, 2 ] results in a revolution around the unit circle, as shown below. 70 Period of Sine and Cosine The values of sin(t + 2) and cos(t + 2) correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result sin(t + 2 n) = sin t and cos(t + 2 n) = cos t for any integer n and real number t. Functions that behave in such a repetitive (or cyclic) manner are called periodic. 71 Period of Sine and Cosine Because 0 and 2𝜋 are in the same position on unit circle (2𝜋 completes a second revolution around a unit circle) the following statements are true: sin(t + 2𝜋n) = sin t cos(t + 2𝜋n) = cos t 72 Period of Sine and Cosine 3 What about the period of tangent? undef 3 1 1 Let's label the unit circle with values of the tangent. (Remember this is just y/x) 3 3 3 3 0 0 3 3 We see that they repeat every so the tangent’s period is . 1 1 3 , 2 2 3 undef tan( + ) = tan 3 3 1 3 73 Reciprocal Functions have the same Period. PERIODIC PROPERTIES sin( + 2) = sin cosec( + 2) = cosec cos( + 2) = cos sec( + 2) = sec tan( + ) = tan cot( + ) = cot 9 tan 1 4 This would have the same value as tan 4 (you can count around on unit circle or subtract the period twice.) 74 Example – Evaluating Sine and Cosine a. Because , you have Subtract multiples of 2𝜋. b. Because , you have Add multiples of 2𝜋. 75 YOUR TURN: Evaluate the trigonometric function using its period as an aid cos5 9 sin 4 8 cos 3 19 sin 6 76 Solution: cos 5𝜋 Subtract out full periods of 2𝜋. 5𝜋 – 2(2𝜋) = 5𝜋 - 4𝜋 = 𝜋 So, cos 5𝜋 = cos 𝜋 = −1 cos 5𝜋 = −1 77 Solution: 9 sin 4 Subtract out full periods of 2𝜋. 9𝜋 9𝜋 8𝜋 𝜋 − 2𝜋 = − = 4 4 4 4 9𝜋 𝜋 2 sin = sin = 4 4 2 9𝜋 2 sin = 4 2 78 Solution: 8𝜋 cos − 3 Add in full periods of 2𝜋. 8𝜋 8𝜋 12𝜋 4𝜋 − + 4𝜋 = − + = 3 3 3 3 8𝜋 4𝜋 1 cos − = cos =− 3 3 2 8𝜋 1 cos − =− 3 2 79 Even and Odd Circular Functions Now let’s look at the unit circle to compare trig functions of positive vs. negative angles. What is cos 3 ? What is cos ? 3 1 2 1 2 Recall that if we put a negative in the function and get the original back (𝑓 −𝑥 = 𝑓(𝑥)) it is an even function. Therefore, cosine is an even function and cos cos 1 3 , 2 2 80 Even and Odd Circular Functions What is sin 3 ? 3 2 3 What is sin ? 2 3 Recall that if we put a negative in the function and get the negative of the function back (𝑓 −𝑥 = −𝑓(𝑥)) it is an odd function. Therefore, sine is an odd function and sin sin 1 3 , 2 2 81 Even and Odd Circular Functions What is tan 3 ? What is tan ? 3 3 3 Again, if we put a negative in the function and get the negative of the function back (𝑓 −𝑥 = −𝑓(𝑥)) it is an odd function. Therefore, tangent is an odd function and tan tan 1 3 , 2 2 82 Even and Odd Circular Functions We know that a function f is even when f(–t) = f(t), and is odd when f(–t) = –f(t). If a function is even, its reciprocal function will be also. If a function is odd its reciprocal will be also. 83 Example: sin 60 what in terms of a positive angle? sin 60 2 sec 3 what in terms of a positive angle? 2 sec 3 84 Your Turn: Use the value of the trigonometric function to evaluate the indicated functions: sin(-t) = 3/8 1) sin t − 3 8 2) csc t − 8 3 85 Domain and Period of Sine and Cosine When evaluating a trigonometric function with a calculator, you need to set the calculator to the desired mode of measurement (degree or radian). Most calculators do not have keys for the cosecant, secant, and cotangent functions. To evaluate these functions, you can use the key with their respective reciprocal functions: sine, cosine, and tangent. 86 Domain and Period of Sine and Cosine For instance, to evaluate csc( /8), use the fact that and enter the following keystroke sequence in radian mode. Display 2.6131259 87 Example 3 – Using a Calculator Function Mode Calculator Keystrokes Display 88 Applications of Circular Functions The phase F of the moon is given by 1 F (t ) (1 cos t) 2 where t is called the phase angle. F(t) gives the fraction of the moon’s face illuminated by the sun. 89 Modeling the Phases of the Moon Evaluate and interpret. (a) F (0) π (b) F 2 (c) F (π) 3π (d) F 2 1 F (t ) (1 cos t) 2 90 Solution: a) b) 1 F (t ) (1 cos t) 2 1 1 F (0) (1 cos0) (11) 0 2 2 π 1 1 1 F (1 cos π ) [1 0] 2 2 2 2 2 new moon first quarter c) F (π) 1 (1 cos π) 1 [1 (1)] 1 full moon d) F 3π 1 (1 cos 3π ) 1 [1 0] 1 2 2 2 2 2 last quarter 2 2 91 Practice Exercises 1. Find the value of the sec 360 ̊ without using a calculator. 2. Find the exact value of the tan 420 ̊ . 5 3. Find the exact value of sin 6 . 4. Find the tan 270 ̊ without using a calculator. 5. Find the exact value of the csc 7 3 . 6. Find the exact value of the cot (-225 ̊ ). 7. Find the exact value of the sin 8. Find the exact value of the cos 13 4 11 6 . . 9. Find the value of the cos(- ) without using a calculator. 10. Find the exact value of the sec 315 ̊ . 92 Key For The Practice Exercises 1. sec 360 ̊ = 1 2. tan 420 ̊ = 3. sin 5 6 3 = 1 2 4. tan 270 ̊ is undefined 5. csc 7 3 2 = 3 2 3 3 6. cot (-225 ̊) = -1 7. sin 8. cos 13 4 11 6 = = 1 2 2 2 3 2 9. cos(- ) = -1 10. sec 315 ̊ = 2 93 Assignment Pg. 274 -276: #1 – 67 odd, 75 – 79 odd 94