The Other Trigonometric Functions

Transcription

1 OpenStax-CNX module: m4974 The Other Trigonometric Functions OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will: Abstract Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of π, π 4, and π 6. Use reference angles to evaluate the trigonometric functions secant, cosecant, tangent, and cotangent. Use properties of even and odd trigonometric functions. Recognize and use fundamental identities. Evaluate trigonometric functions with a calculator. A wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground whose tangent is or less, regardless of its length. A tangent represents a ratio, so this means that for every inch of rise, the ramp must have inches of run. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. We have already dened the sine and cosine functions of an angle. Though sine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of six trigonometric functions. In this section, we will investigate the remaining functions. Finding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent, and Cotangent To dene the remaining functions, we will once again draw a unit circle with a point (x, y) corresponding to an angle of t,as shown in Figure. As with the sine and cosine, we can use the (x, y) coordinates to nd the other functions. Version.7: Feb 4, 05 :46 am

2 OpenStax-CNX module: m4974 Figure The rst function we will dene is the tangent. The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. In Figure, the tangent of angle t is equal to y x, x 0. Because the y-value is equal to the sine of t, and the x-value is equal to the cosine of t, the tangent of angle t can also be dened as sin t cos t, cos t 0.The tangent function is abbreviated as tan. The remaining three functions can all be expressed as reciprocals of functions we have already dened. The secant function is the reciprocal of the cosine function. In Figure, the secant of angle t is equal to cos t = x, x 0. The secant function is abbreviated as sec. The cotangent function is the reciprocal of the tangent function. In Figure, the cotangent of angle t is equal to cos t sin t = x y, y 0. The cotangent function is abbreviated as cot. The cosecant function is the reciprocal of the sine function. In Figure, the cosecant of angle t is equal to sin t = y, y 0. The cosecant function is abbreviated as csc. A General Note: If t is a real number and (x, y) is a point where the terminal side of an angle of t radians intercepts the unit circle, then tan t = y x, x 0 sec t = x, x 0 csc t = y, y 0 cot t = x y, y 0 () Example Finding Trigonometric ( Functions from a Point on the Unit Circle The point ), is on the unit circle, as shown in Figure. Find sin t, cos t, tan t, sec t, csc t, and cot t.

3 OpenStax-CNX module: m4974 Figure Solution Because we know the (x, y) coordinates of the point on the unit circle indicated by angle t, we can use those coordinates to nd the six functions: sin t = y = cos t = x = tan t = y x = sec t = x = csc t = y = cot t = x y = = ) ( = = = = = = ( ) = () Try It: Exercise (Solution on p. 8.) ( The point, ) is on the unit circle, as shown in Figure. Find sin t, cos t, tan t, sec t, csc t, and cot t.

4 OpenStax-CNX module: m Figure Example Finding the Trigonometric Functions of an Angle Find sin t, cos t, tan t, sec t, csc t, and cot t when t = π 6. Solution We have previously used the properties of equilateral triangles to demonstrate that sin π 6 = and cos π 6 =. We can use these values and the denitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to nd the remaining function values. tan π 6 = sin π 6 cos π 6 = = = () sec π 6 = cos π 6 = = = (4)

5 OpenStax-CNX module: m csc π 6 = sin π 6 = = (5) cot π 6 = cos π 6 = sin π 6 = (6) Try It: Exercise 4 (Solution on p. 8.) Find sin t, cos t, tan t, sec t, csc t, and cot t when t = π. Because we know the sine and cosine values for the common rst-quadrant angles, we can nd the other function values for those angles as well by setting x equal to the cosine and y equal to the sine and then using the denitions of tangent, secant, cosecant, and cotangent. The results are shown in Table. π Angle 0 6, or 0 π 4, or 45 π, or 60 π, or 90 Cosine 0 Sine 0 Tangent 0 Undened Secant Undened Cosecant Undened Cotangent Undened 0 Table Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent We can evaluate trigonometric functions of angles outside the rst quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x- and y-values in the original quadrant. Figure 4 shows which functions are positive in which quadrant. To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase A Smart Trig Class. Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is A, all of the six trigonometric functions are positive. In quadrant II, Smart, only sine and its reciprocal function, cosecant, are positive. In quadrant III, Trig, only tangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, Class, only cosine and its reciprocal function, secant, are positive.

6 OpenStax-CNX module: m Figure 4 How To: Given an angle not in the rst quadrant, use reference angles to nd all six trigonometric functions..measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle..evaluate the function at the reference angle..observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative. Example Using Reference Angles to Find Trigonometric Functions Use reference angles to nd all six trigonometric functions of 5π 6. Solution The angle between this angle's terminal side and the x-axis is π 6, so that is the reference angle. Since 5π 6 is in the third quadrant, where both x and y are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.

7 OpenStax-CNX module: m cos ( ) 5π 6 =, sin ( ) 5π 6 =, tan ( ) 5π 6 = sec ( ) 5π 6 =, csc ( ) ( ) (7) 5π 6 =, cot 5π 6 = Try It: Exercise 6 (Solution on p. 8.) Use reference angles to nd all six trigonometric functions of 7π 4. Using Even and Odd Trigonometric Functions To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important dierence among the functions in this regard. Consider the function f (x) = x, shown in Figure 5. The graph of the function is symmetrical about the y-axis. All along the curve, any two points with opposite x-values have the same function value. This matches the result of calculation: (4) = ( 4),( 5) = (5), and so on. So f (x) = x is an even function, a function such that two inputs that are opposites have the same output. That means f ( x) = f (x). Figure 5: The function f (x) = x is an even function.

8 OpenStax-CNX module: m Now consider the function f (x) = x, shown in Figure 6. The graph is not symmetrical about the y-axis. All along the graph, any two points with opposite x-values also have opposite y-values. So f (x) = x is an odd function, one such that two inputs that are opposites have outputs that are also opposites. That means f ( x) = f (x).

9 OpenStax-CNX module: m Figure 6: The function f (x) = x is an odd function.

10 OpenStax-CNX module: m We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure 7. The sine of the positive angle is y. The sine of the negative angle is y. The sine function, then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table. Figure 7 sin t = y sin ( t) = y sin t sin ( t) sec t = x sec ( t) = x sec t = sec ( t) cos t = x cos ( t) = x cos t = cos ( t) csc t = y csc ( t) = y csc t csc ( t) Table tan (t) = y x tan ( t) = y x tan t tan ( t) cot t = x y cot ( t) = x y cot t cot ( t)

11 OpenStax-CNX module: m4974 A General Note: An even function is one in which f ( x) = f (x). An odd function is one in which f ( x) = f (x). Cosine and secant are even: Sine, tangent, cosecant, and cotangent are odd: cos ( t) = cos t sec ( t) = sec t (8) sin ( t) = sin t tan ( t) = tan t csc ( t) = csc t cot ( t) = cot t (9) Example 4 Using Even and Odd Properties of Trigonometric Functions If the secant of angle t is, what is the secant of t? Solution Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is, the secant of t is also. Try It: Exercise 8 (Solution on p. 8.) If the cotangent of angle t is, what is the cotangent of t? 4 Recognizing and Using Fundamental Identities We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are dened. Usually, identities can be derived from denitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the denitions of sine and cosine. A General Note: We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships: tan t = sin t cos t sec t = cos t (0) ()

12 OpenStax-CNX module: m4974 csc t = sin t cot t = tan t = cos t sin t Example 5 Using Identities to Evaluate Trigonometric Functions () () a. Given sin (45 ) =, cos (45 ) =, evaluate tan (45 ). b. Given sin ( ) 5π 6 =, cos ( ) 5π 6 =, evaluate sec ( ) 5π 6. Solution Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions. a. b. tan (45 ) = sin(45 ) = = cos(45 ) sec ( ) 5π 6 = cos( 5π 6 ) = = = = (4) (5) Try It: Exercise 0 (Solution on p. 8.) Evaluate csc ( ) 7π 6. Example 6 Using Identities to Simplify Trigonometric Expressions Simplify sec t tan t. Solution We can simplify this by rewriting both functions in terms of sine and cosine. (6) By showing that sec t tan t can be simplied to csc t,we have, in fact, established a new identity.

13 OpenStax-CNX module: m4974 sec t = csc t (7) tan t Try It: Exercise (Solution on p. 8.) Simplify tan t (cos t). 4. Alternate Forms of the Pythagorean Identity We can use these fundamental identities to derive alternative forms of the Pythagorean Identity, cos t + sin t =. One form is obtained by dividing both sides by cos t : cos t cos t + sin t cos t = cos t + tan t = sec t (8) The other form is obtained by dividing both sides by sin t : cos t sin t + sin t sin t = sin t cot t + = csc t (9) A General Note: + tan t = sec t (0) cot t + = csc t () Example 7 Using Identities to Relate Trigonometric Functions If cos (t) = and t is in quadrant IV, as shown in Figure 8, nd the values of the other ve trigonometric functions.

14 OpenStax-CNX module: m Figure 8 Solution We can nd the sine using the Pythagorean Identity, cos t +sin t =, and the remaining functions by relating them to sine and cosine. ( ) + sin t = sin t = ( sin t = sin t = 5 69 sin t = ± ) 5 69 sin t = ± 5 69 () sin t = ± 5 The sign of the sine depends on the y-values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y-values are negative, its sine is negative, 5.

15 OpenStax-CNX module: m The remaining functions can be calculated using identities relating them to sine and cosine. tan t = sin t cos t = 5 = 5 sec t = cos t = csc t = sin t = 5 cot t = tan t = 5 = = 5 = 5 () Try It: Exercise 4 (Solution on p. 8.) If sec (t) = 7 8 and 0 < t < π, nd the values of the other ve functions. As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or π, will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs. Other functions can also be periodic. For example, the lengths of months repeat every four years. If x represents the length time, measured in years, and f (x) represents the number of days in February, then f (x + 4) = f (x). This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satises this condition and is called the period. A period is the shortest interval over which a function completes one full cyclein this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days. A General Note: The period P of a repeating function f is the number representing the interval such that f (x + P ) = f (x) for any value of x. The period of the cosine, sine, secant, and cosecant functions is π. The period of the tangent and cotangent functions is π. Example 8 Finding the Values of Trigonometric Functions Find the values of the six trigonometric functions of angle t based on Figure 9.

16 OpenStax-CNX module: m Figure 9 Solution sin t = y = cos t = x = tan t = sint cost = sec t = cost = csc t = sint = = = = cot t = tant = = (4)

17 OpenStax-CNX module: m Try It: Exercise 6 (Solution on p. 8.) Find the values of the six trigonometric functions of angle t based on Figure 0. Figure 0 Example 9 Finding the Value of Trigonometric Functions If sin (t) = and cos (t) =,nd sec (t), csc (t), tan (t), cot (t). Solution

18 OpenStax-CNX module: m sec t = cos t = csc t = sin t = tan t = sin t cos t = = = cot t = tan t = = (5) Try It: Exercise 8 (Solution on p. 8.) If sin (t) = and cos (t) =,nd sec (t), csc (t), tan (t), and cot (t). 5 Evaluating Trigonometric Functions with a Calculator We have learned how to evaluate the six trigonometric functions for the common rst-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientic or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, conrm the correct mode is chosen before making a calculation. Evaluating a tangent function with a scientic calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent. If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor π 80 to convert the degrees to radians. To nd the secant of 0, we could press or (for a scientic calculator): 0 π COS (6) 80 (for a graphing calculator): cos ( ) 0π (7) 80 How To: cosecant. Given an angle measure in radians, use a scientic calculator to nd the.if the calculator has degree mode and radian mode, set it to radian mode..enter: /.Enter the value of the angle inside parentheses. 4.Press the SIN key. 5.Press the = key.

19 OpenStax-CNX module: m How To: Given an angle measure in radians, use a graphing utility/calculator to nd the cosecant..if the graphing utility has degree mode and radian mode, set it to radian mode..enter: /.Press the SIN key. 4.Enter the value of the angle inside parentheses. 5.Press the ENTER key. Example 0 Evaluating the Secant Using Technology Evaluate the cosecant of 5π 7. Solution For a scientic calculator, enter information as follows: / ( 5 π / 7 ) SIN = (8) csc ( ) 5π.79 (9) 7 Try It: Exercise 0 (Solution on p. 8.) Evaluate the cotangent of π 8. Media: Access these online resources for additional instruction and practice with other trigonometric functions. Determining Trig Function Values More Examples of Determining Trig Functions Pythagorean Identities Trig Functions on a Calculator

20 OpenStax-CNX module: m Key Equations Tangent function tan t = sint cost Secant function sec t = cost Cosecant function csc t = sint Cotangent function cot t = tan t = cos t sin t Table 7 Key Concepts The tangent of an angle is the ratio of the y-value to the x-value of the corresponding point on the unit circle. The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function. The six trigonometric functions can be found from a point on the unit circle. See Example. Trigonometric functions can also be found from an angle. See Example. Trigonometric functions of angles outside the rst quadrant can be determined using reference angles. See Example. A function is said to be even if f ( x) = f (x) and odd if f ( x) = f (x). Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd. Even and odd properties can be used to evaluate trigonometric functions. See Example 4. The Pythagorean Identity makes it possible to nd a cosine from a sine or a sine from a cosine. Identities can be used to evaluate trigonometric functions. See Example 5 and Example 6. Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See Example 7. The trigonometric functions repeat at regular intervals. The period P of a repeating function f is the smallest interval such that f (x + P ) = f (x) for any value of x. The values of trigonometric functions of special angles can be found by mathematical analysis. To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See Example 0. 8 Section Exercises 8. Verbal Exercise (Solution on p. 8.) On an interval of [0, π), can the sine and cosine values of a radian measure ever be equal? If so, where? Exercise What would you estimate the cosine of π degrees to be? Explain your reasoning. Exercise (Solution on p. 8.) For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle? Exercise 4 Describe the secant function.

21 OpenStax-CNX module: m4974 Exercise 5 (Solution on p. 8.) Tangent and cotangent have a period of π. What does this tell us about the output of these functions? 8. Algebraic For the following exercises, nd the exact value of each expression. Exercise 6 tan π 6 Exercise 7 (Solution on p. 8.) sec π 6 Exercise 8 csc π 6 Exercise 9 (Solution on p. 8.) cot π 6 Exercise 0 tan π 4 Exercise (Solution on p. 8.) sec π 4 Exercise csc π 4 Exercise (Solution on p. 8.) cot π 4 Exercise 4 tan π Exercise 5 (Solution on p. 8.) sec π Exercise 6 csc π Exercise 7 (Solution on p. 9.) cot π For the following exercises, use reference angles to evaluate the expression. Exercise 8 tan 5π 6 Exercise 9 (Solution on p. 9.) sec 7π 6 Exercise 40 csc π 6 Exercise 4 (Solution on p. 9.) cot π 6 Exercise 4 tan 7π 4 Exercise 4 (Solution on p. 9.) sec π 4

22 OpenStax-CNX module: m4974 Exercise 44 csc 5π 4 Exercise 45 (Solution on p. 9.) cot π 4 Exercise 46 tan 8π Exercise 47 (Solution on p. 9.) sec 4π Exercise 48 csc π Exercise 49 (Solution on p. 9.) cot 5π Exercise 50 tan 5 Exercise 5 (Solution on p. 9.) sec 00 Exercise 5 csc 50 Exercise 5 (Solution on p. 9.) cot 40 Exercise 54 tan 0 Exercise 55 (Solution on p. 9.) sec 0 Exercise 56 csc 0 Exercise 57 (Solution on p. 9.) cot 5 Exercise 58 If sin t = 4, and t is in quadrant II, nd cos t, sec t, csc t, tan t, cot t. Exercise 59 (Solution on p. 9.) If cos t =, and t is in quadrant III, nd sin t, sec t, csc t, tan t, cot t. Exercise 60 If tan t = 5, and 0 t < π, nd sin t, cos t, sec t, csc t, and cot t. Exercise 6 (Solution on p. 9.) If sin t = and cos t =, nd sec t, csc t, tan t, and cot t. Exercise 6 If sin cos sec 40, csc 40, tan 40, and cot 40. Exercise 6 (Solution on p. 9.) If sin t =, what is the sin ( t)? Exercise 64 If cos t =, what is the cos ( t)? Exercise 65 (Solution on p. 9.) If sec t =., what is the sec ( t)?

23 OpenStax-CNX module: m4974 Exercise 66 If csc t = 0.4, what is the csc ( t)? Exercise 67 (Solution on p. 9.) If tan t =.4, what is the tan ( t)? Exercise 68 If cot t = 9., what is the cot ( t)? 8. Graphical For the following exercises, use the angle in the unit circle to nd the value of the each of the six trigonometric functions. Exercise 69 (Solution on p. 9.)

24 OpenStax-CNX module: m Exercise 70

25 OpenStax-CNX module: m Exercise 7 (Solution on p. 9.) 8.4 Technology For the following exercises, use a graphing calculator to evaluate. Exercise 7 csc 5π 9 Exercise 7 (Solution on p. 9.) cot 4π 7 Exercise 74 sec π 0 Exercise 75 (Solution on p. 9.) tan 5π 8 Exercise 76 sec π 4 Exercise 77 (Solution on p. 9.) csc π 4

26 OpenStax-CNX module: m Exercise 78 tan 98 Exercise 79 (Solution on p. 9.) cot Exercise 80 cot 40 Exercise 8 (Solution on p. 9.) sec Extensions For the following exercises, use identities to evaluate the expression. Exercise 8 If tan (t).7, and sin (t) 0.94, nd cos (t). Exercise 8 (Solution on p. 9.) If tan (t)., and cos (t) 0.6, nd sin (t). Exercise 84 If csc (t)., and cos (t) 0.95, nd tan (t). Exercise 85 (Solution on p. 0.) If cot (t) 0.58, and cos (t) 0.5, nd csc (t). Exercise 86 Determine whether the function f (x) = sin x cos x is even, odd, or neither. Exercise 87 (Solution on p. 0.) Determine whether the function f (x) = sin x cos x + sec x is even, odd, or neither. Exercise 88 Determine whether the function f (x) = sin x cos x is even, odd, or neither. Exercise 89 (Solution on p. 0.) Determine whether the function f (x) = csc x + sec x is even, odd, or neither. For the following exercises, use identities to simplify the expression. Exercise 90 csc t tan t Exercise 9 (Solution on p. 0.) sec t csc t 8.6 Real-World Applications Exercise 9 The amount of sunlight in a certain city can be modeled by the function h = 5cos ( 600 d), where h represents the hours of sunlight, and d is the day of the year. Use the equation to nd how many hours of sunlight there are on February 0, the 4 nd day of the year. State the period of the function. Exercise 9 (Solution on p. 0.) The amount of sunlight in a certain city can be modeled by the function h = 6cos ( 500 d), where h represents the hours of sunlight, and d is the day of the year. Use the equation to nd how many

27 OpenStax-CNX module: m hours of sunlight there are on September 4, the 67 th day of the year. State the period of the function. Exercise 94 The equation P = 0sin (πt) + 00 models the blood pressure, P, where t represents time in seconds. (a) Find the blood pressure after 5 seconds. (b) What are the maximum and minimum blood pressures? Exercise 95 (Solution on p. 0.) The height of a piston, h, in inches, can be modeled by the equation y = cos x+6, where x represents the crank angle. Find the height of the piston when the crank angle is 55. Exercise 96 The height of a piston, h,in inches, can be modeled by the equation y = cos x+5,where x represents the crank angle. Find the height of the piston when the crank angle is 55.

28 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. ) sin t =, cos t =, tan t =, sec t =, csc t =, cot t = Solution to Exercise (p. 5) sin π = cos π = tan π = sec π = csc π = cot π = Solution to Exercise (p. 7) sin ( 7π 4 sec ( 7π 4 ) =, cos ( ) 7π 4 =, tan ( ) 7π 4 =, ) ( =, csc 7π ) ( 4 =, cot 7π ) 4 = Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) sin t Solution to Exercise (p. 5) cos t = , sin t = 7, tan t = 8 csc t = 7 5, cot t = 8 5 Solution to Exercise (p. 7) sin t =, cos t = 0, tan t = Undened sec t = Undened,csc t =, cot t = 0 Solution to Exercise (p. 8) sec t =, csc t =, tan t =, cot t = Solution to Exercise (p. 9).44 Solution to Exercise (p. 0) Yes, when the reference angle is π 4 and the terminal side of the angle is in quadrants I and III. Thus, at x = π 4, 5π 4, the sine and cosine values are equal. Solution to Exercise (p. 0) Substitute the sine of the angle in for y in the Pythagorean Theorem x + y =. Solve for x and take the negative solution. Solution to Exercise (p. ) The outputs of tangent and cotangent will repeat every π units. Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. )

29 OpenStax-CNX module: m Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Solution to Exercise (p. ) Ifsin t =, sec t =, csc t = 4, tan t =, cot t = 4 Solution to Exercise (p. ) sec t =, csc t =, tan t =, cot t = Solution to Exercise (p. ) Solution to Exercise (p. ). Solution to Exercise (p. ).4 Solution to Exercise (p. ) sin t =, cos t =, tan t =, cot t =, sec t =, csc t = Solution to Exercise (p. 5) sin t =, cos t =, tan t =, cot t =, sec t =, csc t = Solution to Exercise (p. 5) 0.8 Solution to Exercise (p. 5).44 Solution to Exercise (p. 5).44 Solution to Exercise (p. 6).540 Solution to Exercise (p. 6).556

30 OpenStax-CNX module: m Solution to Exercise (p. 6) sin (t) 0.79 Solution to Exercise (p. 6) csct.6 Solution to Exercise (p. 6) even Solution to Exercise (p. 6) even Solution to Exercise (p. 6) sin t cos t = tan t Solution to Exercise (p. 6).77 hours, period: 000π Solution to Exercise (p. 7) 7.7 inches Glossary Denition : cosecant the reciprocal of the sine function: on the unit circle, csc t = y, y 0 Denition : cotangent the reciprocal of the tangent function: on the unit circle, cot t = x y, y 0 Denition : identities statements that are true for all values of the input on which they are dened Denition 4: period the smallest interval P of a repeating function f such that f (x + P ) = f (x) Denition 5: secant the reciprocal of the cosine function: on the unit circle, sec t = x, x 0 Denition 6: tangent the quotient of the sine and cosine: on the unit circle, tan t = y x, x 0