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OpenStax-CNX module: m49374 1 The Other Trigonometric Functions ∗ OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 † Abstract In this section, you will: • • Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of π π , , and π6 . 3 4 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 1 12 or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of rise, the ramp must have 12 inches of run. Trigonometric functions allow us with the ground whose tangent is 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. 1 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) an angle of t,as shown in Figure 1. As with the sine and cosine, we can use the coordinates to nd the other functions. ∗ Version 1.7: Feb 24, 2015 11:46 am -0600 † http://creativecommons.org/licenses/by/4.0/ http://https://legacy.cnx.org/content/m49374/1.7/ (x, y) corresponding to OpenStax-CNX module: m49374 2 Figure 1 The rst function we will dene is the tangent. The tangent to the x -value of the corresponding point on the unit circle. of an angle is the ratio of the y -value In Figure 1, the tangent of angle t is equal y x , x 6= 0. Because the y -value is equal to the sine of t, and the x -value is equal to the cosine of t, the sin t tangent of angle t can also be dened as cos t , cos t 6= 0.The tangent function is abbreviated as tan. The remaining three functions can all be expressed as reciprocals of functions we have already dened. to • The • secant function is the reciprocal of the cosine function. 1 cos t The to = x1 , x 6= 0. The cotangent function is the reciprocal of the tangent function. equal to • The cos t sin t = xy , y 6= 0. The 1 sin t = y1 , y 6= 0. The cosecant equal to t If t In Figure 1, the cotangent of angle t is cotangent function is abbreviated as cot. function is the reciprocal of the sine function. A General Note: of In Figure 1, the secant of angle t is equal secant function is abbreviated as sec. In Figure 1, the cosecant of angle t is cosecant function is abbreviated as csc. is a real number and (x, y) is a point where the terminal side of an angle radians intercepts the unit circle, then tan t = xy , x 6= 0 sec t = x1 , x 6= 0 csc t = y1 , y 6= 0 (1) cot t = xy , y 6= 0 Example 1 Finding Trigonometric Functions from a Point on the Unit Circle √ The point − 3 1 2 , 2 is on the unit circle, as shown in Figure 2. Find sin t, cos t, tan t, sec t, csc t, and cot t. http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 3 Figure 2 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 = 1 2 cos t = x = − tan t = y x = sec t = 1 x = csc t = 1 y = cot t = x y = √ 3 2 1 2 √ − 23 1 √ − 3 2 1 1 2 √ − √23 = − √13 = − 33 √ (2) = − √23 = − 2 3 3 =2 1 2 √ 3 2 1 2 − = √ =− 3 2 2 1 √ =− 3 Try It: Exercise 2 The point (Solution on p. 28.) √ 2 2 , − is on the unit circle, as shown in Figure 3. Find sin t, cos t, tan t, sec t, csc t, 2 2 √ and cot t. http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 4 Figure 3 Example 2 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 = 3 1 π 2 and cos 6 = 2 . 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 = sec π6 = = http://https://legacy.cnx.org/content/m49374/1.7/ = sin cos 1 2 √ 3 2 = 1 cos π 6 1 √ 3 2 π 6 π 6 = √1 3 √2 3 √ = = (3) 3 3 √ 2 3 3 (4) OpenStax-CNX module: m49374 5 csc π 1 1 = = 1 =2 6 sin π6 2 cot π6 = = cos π 6 π sin √ 6 3 2 1 2 = √ (5) (6) 3 Try It: Exercise 4 (Solution on p. 28.) Find sin t, cos t, tan t, sec t, csc t, and cot t when t = π 3. 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 1. Angle Cosine Sine Tangent Secant Cosecant Cotangent 1 π 6, √ 3 2 1 2 √ 3 3 √ 2 3 3 Undened 2 0 1 0 0 √ Undened or 30 ◦ π 4, √ 2 2 √ 2 2 1 √ √ 3 2 2 1 or 45 ◦ π 3, 1 2 √ 3 2 √ or 60 ◦ π 2, or 90 ◦ 0 1 3 Undened 2 Undened √ 2 3 3 √ 3 3 1 0 Table 1 2 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 all of the six trigonometric functions are positive. function, cosecant, are positive. In quadrant III, are positive. Finally, in quadrant IV, Class, http://https://legacy.cnx.org/content/m49374/1.7/ In quadrant II, Smart, only sine A, and its reciprocal Trig, only tangent and its reciprocal function, cotangent, cosine and its reciprocal function, secant, are positive. only OpenStax-CNX module: m49374 6 Figure 4 Given an angle not in the rst quadrant, use reference angles to nd all six trigonometric functions. How To: 1.Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle. 2.Evaluate the function at the reference angle. 3.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 3 Using Reference Angles to Find Trigonometric Functions Use reference angles to nd all six trigonometric functions of − Solution 5π 6 . π 6 , so that is the reference angle. 5π Since − is in the third quadrant, where both x and y are negative, cosine, sine, secant, and cosecant 6 will be negative, while tangent and cotangent will be positive. The angle between this angle's terminal side and the x -axis is http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 7 √ √3 cos − 5π = − 23 , sin − 5π = − 21 , tan − 5π = 3 6 6 6 √ √ 5π sec − = − 2 3 3 , csc − 5π = −2, cot − 5π = 3 6 6 6 (7) Try It: Exercise 6 (Solution on p. 28.) Use reference angles to nd all six trigonometric functions of − 7π 4 . 3 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) = x2 , 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 2 matches the result of calculation: (4) 2 2 2 = (−4) ,(−5) = (5) , and so on. So f (x) = x2 is a function such that two inputs that are opposites have the same output. That means f Figure 5: The function f (x) = x2 is an even function. http://https://legacy.cnx.org/content/m49374/1.7/ an even function, (−x) = f (x) . OpenStax-CNX module: m49374 Now consider the function f 8 (x) = x3 , 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. an So f (x) = x3 is odd function, one such that two inputs that are opposites have outputs that are also opposites. means f (−x) = −f (x) . http://https://legacy.cnx.org/content/m49374/1.7/ That OpenStax-CNX module: m49374 9 http://https://legacy.cnx.org/content/m49374/1.7/ 3 The function is an odd function. Figure 6: f (x) = x OpenStax-CNX module: m49374 10 unit circle with a positive We can test whether a trigonometric function is even or odd by drawing a and a negative angle, as in Figure 7. The sine of the positive angle is y. The sine of the negative angle is The sine function, −y. then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in Table 2. Figure 7 sin t = y sin (−t) = −y sin t 6= sin (−t) sec t = sec (−t) = 1 x 1 x sec t = sec (−t) cos t =x cos (−t) = x cos t = cos (−t) csc t = csc (−t) = 1 y 1 −y csc t 6= csc (−t) Table 2 http://https://legacy.cnx.org/content/m49374/1.7/ tan (t) = y x tan (−t) = − xy tan t 6= tan (−t) cot t = cot (−t) = x y x −y cot t 6= cot (−t) OpenStax-CNX module: m49374 A General Note: An 11 An even function is one in which f (−x) = f (x) . odd function is one in which f (−x) = −f (x) . Cosine and secant are even: cos (−t) = cos t (8) sec (−t) = sec t Sine, tangent, cosecant, and cotangent are odd: sin (−t) = −sin t tan (−t) = −tan t (9) csc (−t) = −csc t cot (−t) = −cot t Example 4 Using Even and Odd Properties of Trigonometric Functions If the secant of angle t is 2, 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 2, the secant of − t is also 2. Try It: Exercise 8 If the cotangent of angle t is √ (Solution on p. 28.) 3, 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: = sin t cos t (10) sec t = 1 cos t (11) tan t http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 12 1 sin t csc t = cot t = 1 = tan t (12) cos t (13) sin t Example 5 Using Identities to Evaluate Trigonometric Functions √ √ sin (45 ◦ ) = 22 , cos (45 ◦ ) = 22 , evaluate tan (45 ◦ ) . √ 1 5π Given sin = 2 , cos 5π = − 23 , evaluate sec 5π 6 6 6 . a. Given b. Solution Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions. a. tan (45 ◦ ) = = sin(45 ◦ ) ◦) √ cos(45 2 2 √ 2 2 (14) =1 b. sec 5π 6 = 1 cos( 5π 6 ) 1√ − 23 √ = −21 3 −2 =√ 3 √ = −233 = (15) Try It: Exercise 10 Evaluate csc (Solution on p. 28.) 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. (16) By showing that sec t tan t can be simplied to csc t,we have, in fact, established a new identity. http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 13 sec t = csc t tan t (17) Try It: Exercise 12 (Solution on p. 28.) Simplify tan t (cos t) . 4.1 Alternate Forms of the Pythagorean Identity We can use these fundamental identities to derive alternative forms of the 2 sin t = 1. One form is obtained by dividing both sides by cos cos2 t cos2 t + sin2 t cos2 t 2 = 2 Pythagorean Identity, cos2 t + t: 1 cos2 t 2 (18) 1 + tan t = sec t The other form is obtained by dividing both sides by cos2 t sin2 t + sin2 t : sin2 t sin2 t = 1 sin2 t 2 (19) cot2 t + 1 = csc t A General Note: 1 + tan2 t = sec2 t (20) cot2 t + 1 = csc2 t (21) Example 7 Using Identities to Relate Trigonometric Functions 12 13 and t is in quadrant IV, as shown in Figure 8, nd the values of the other ve trigonometric functions. If cos (t) = http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 14 Figure 8 Solution We can nd the sine using the Pythagorean Identity, cos 2 t + sin2 t = 1, and the remaining functions by relating them to sine and cosine. 12 2 13 + sin2 t = 1 sin2 t = 1 − sin2 t = 1 − sin2 t = sin t = sin t = sin t = 12 2 13 144 169 25 169q (22) 25 ± 169 √ 25 ± √169 5 ± 13 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, http://https://legacy.cnx.org/content/m49374/1.7/ − 5 13 . OpenStax-CNX module: m49374 15 The remaining functions can be calculated using identities relating them to sine and cosine. tan t = sec t = sin t cos t 1 cos t = = 5 − 13 5 = − 12 12 13 1 12 13 1 sin t csc t = = −15 13 1 1 cot t = tan = 5 t − 12 = = 13 12 (23) − 13 5 = − 12 5 Try It: Exercise 14 If sec (t) = (Solution on p. 28.) − 17 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 2π, 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 then f (x + 4) = f (x) . (x) represents the number of days in February, 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: such that f The period P of a repeating function f is the number representing the interval (x + P ) = f (x) for any value of x. The period of the cosine, sine, secant, and cosecant functions is 2π. 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 http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 16 Figure 9 Solution √ 3 2 cos t = x = − 12 √ √ − 3 sint tan t = cost = − 12 = 3 2 1 sec t = cost = −11 = −2 2 √ 1 csc t = sint = 1√3 = − 2 3 3 − 2 √ 1 cot t = tant = √13 = 33 sin t = y = − http://https://legacy.cnx.org/content/m49374/1.7/ (24) OpenStax-CNX module: m49374 17 Try It: Exercise 16 (Solution on p. 28.) . Find the values of the six trigonometric functions of angle t based on Figure 10 Figure 10 Example 9 Finding the√ Value of Trigonometric Functions If sin (t) =− 3 2 and cos (t) = 21 ,nd sec (t) , csc (t) , tan (t) , Solution http://https://legacy.cnx.org/content/m49374/1.7/ cot (t) . OpenStax-CNX module: m49374 18 sec t = csc t = 1 cos t 1 sin t = = tan t = sin t cos t = cot t = 1 tan t = 1 1 2 =2 1√ 3 − √ 2 − 23 1 2 1 √ − 3 − √ 2 3 3 (25) √ =− 3 √ =− 3 3 Try It: Exercise 18 If sin (t) = 2 2 and cos (t) (Solution on p. 28.) √ √ = 2 2 ,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 of 30 ◦ , π 180 to convert the degrees to radians. To nd the secant we could press (for a scientic calculator): 1 π COS 30 × 180 (26) or (for a graphing calculator): How To: cosecant. 1 cos 30π 180 Given an angle measure in radians, use a scientic calculator to nd the 1.If the calculator has degree mode and radian mode, set it to radian mode. 2.Enter: 1 / 3.Enter the value of the angle inside parentheses. 4.Press the SIN key. 5.Press the = key. http://https://legacy.cnx.org/content/m49374/1.7/ (27) OpenStax-CNX module: m49374 19 Given an angle measure in radians, use a graphing utility/calculator to nd the cosecant. How To: 1.If the graphing utility has degree mode and radian mode, set it to radian mode. 2.Enter: 1 / 3.Press the SIN key. 4.Enter the value of the angle inside parentheses. 5.Press the ENTER key. Example 10 Evaluating the Secant Using Technology Evaluate the cosecant of 5π 7 . Solution For a scientic calculator, enter information as follows: 1 / ( 5 × π csc 5π 7 / 7 ) SIN = (28) ≈ 1.279 (29) Try It: Exercise 20 (Solution on p. 28.) Evaluate the cotangent of Media: − π 8. Access these online resources for additional instruction and practice with other trigono- metric functions. • • • • Determining Trig Function Values 1 More Examples of Determining Trig Functions Pythagorean Identities 3 Trig Functions on a Calculator 4 1 http://openstaxcollege.org/l/trigfuncval 2 http://openstaxcollege.org/l/moretrigfun 3 http://openstaxcollege.org/l/pythagiden 4 http://openstaxcollege.org/l/trigcalc http://https://legacy.cnx.org/content/m49374/1.7/ 2 OpenStax-CNX module: m49374 20 6 Key Equations Tangent function tan t = Secant function sec t = Cosecant function csc t = Cotangent function cot t = sint cost 1 cost 1 sint 1 tan t = cos t sin t Table 3 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 1. Trigonometric functions can also be found from an angle. See Example 2. Trigonometric functions of angles outside the rst quadrant can be determined using reference angles. See Example 3. • • • • • • 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 10. 8 Section Exercises 8.1 Verbal Exercise 21 On an interval of (Solution on p. 28.) [0, 2π) , can the sine and cosine values of a radian measure ever be equal? If so, where? Exercise 22 What would you estimate the cosine of π degrees to be? Explain your reasoning. Exercise 23 (Solution on p. 28.) For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle? Exercise 24 Describe the secant function. http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 Exercise 25 21 (Solution on p. 28.) Tangent and cotangent have a period of π. What does this tell us about the output of these functions? 8.2 Algebraic For the following exercises, nd the exact value of each expression. Exercise 26 tan π6 Exercise 27 (Solution on p. 28.) sec π6 Exercise 28 csc π6 Exercise 29 (Solution on p. 28.) cot π6 Exercise 30 tan π4 Exercise 31 (Solution on p. 28.) sec π4 Exercise 32 csc π4 Exercise 33 (Solution on p. 28.) cot π4 Exercise 34 tan π3 Exercise 35 (Solution on p. 28.) sec π3 Exercise 36 csc π3 Exercise 37 (Solution on p. 29.) cot π3 For the following exercises, use reference angles to evaluate the expression. Exercise 38 tan 5π 6 Exercise 39 (Solution on p. 29.) sec 7π 6 Exercise 40 csc 11π 6 Exercise 41 (Solution on p. 29.) cot 13π 6 Exercise 42 tan 7π 4 Exercise 43 sec 3π 4 http://https://legacy.cnx.org/content/m49374/1.7/ (Solution on p. 29.) OpenStax-CNX module: m49374 22 Exercise 44 csc 5π 4 Exercise 45 (Solution on p. 29.) cot 11π 4 Exercise 46 tan 8π 3 Exercise 47 (Solution on p. 29.) sec 4π 3 Exercise 48 csc 2π 3 Exercise 49 (Solution on p. 29.) cot 5π 3 Exercise 50 tan 225 ◦ Exercise 51 (Solution on p. 29.) sec 300 ◦ Exercise 52 csc 150 ◦ Exercise 53 (Solution on p. 29.) cot 240 ◦ Exercise 54 tan 330 ◦ Exercise 55 (Solution on p. 29.) sec 120 ◦ Exercise 56 csc 210 ◦ Exercise 57 (Solution on p. 29.) cot 315 ◦ Exercise 58 If sin t = 43 , and t is in quadrant II, nd cos t, sec t, csc t, tan t, cot t. Exercise 59 If cos t = − 13 , (Solution on p. 29.) and t is in quadrant III, nd sin t, sec t, csc t, tan t, cot t. Exercise 60 If tan t = 12 5 , and 0 Exercise √61 If sin t = ◦ π 2 , nd sin t, cos t, sec t, csc t, and = 12 , nd sec t, csc t, tan t, and ≈ 0.643 cos 40 ◦ ≈ 0.766 sec 40 Exercise √63 If sin t = cot t. (Solution on p. 29.) 3 2 and cos t Exercise 62 If sin 40 ≤t< ◦ cot t. , csc 40 ◦ , tan 40 ◦ , and cot 40 ◦ . (Solution on p. 29.) 2 2 , what is the sin (−t)? Exercise 64 If cos t = 12 , what is the cos (−t)? Exercise 65 If sec t = 3.1, (Solution on p. 29.) what is the sec (−t)? http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 23 Exercise 66 If csc t = 0.34, what is the csc (−t)? Exercise 67 If tan t (Solution on p. 29.) = −1.4, what is the tan (−t)? Exercise 68 If cot t = 9.23, what is the cot (−t)? 8.3 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 http://https://legacy.cnx.org/content/m49374/1.7/ (Solution on p. 29.) OpenStax-CNX module: m49374 Exercise 70 http://https://legacy.cnx.org/content/m49374/1.7/ 24 OpenStax-CNX module: m49374 Exercise 71 25 (Solution on p. 29.) 8.4 Technology For the following exercises, use a graphing calculator to evaluate. Exercise 72 csc 5π 9 Exercise 73 (Solution on p. 29.) cot 4π 7 Exercise 74 π sec 10 Exercise 75 (Solution on p. 29.) tan 5π 8 Exercise 76 sec 3π 4 Exercise 77 csc π4 http://https://legacy.cnx.org/content/m49374/1.7/ (Solution on p. 29.) OpenStax-CNX module: m49374 26 Exercise 78 tan 98 ◦ Exercise 79 (Solution on p. 29.) cot 33 ◦ Exercise 80 cot 140 ◦ Exercise 81 (Solution on p. 29.) sec 310 ◦ 8.5 Extensions For the following exercises, use identities to evaluate the expression. Exercise 82 If tan (t) ≈ 2.7, and sin (t) ≈ 0.94, nd cos (t) . and cos (t) ≈ 0.61, nd sin (t) . and cos (t) ≈ 0.95, nd tan (t) . ≈ 0.5, nd csc (t) . Exercise 83 If tan (t) ≈ 1.3, (Solution on p. 29.) Exercise 84 If csc (t) ≈ 3.2, Exercise 85 If cot (t) ≈ 0.58, (Solution on p. 30.) and cos (t) Exercise 86 Determine whether the function f Exercise 87 Determine whether the function Exercise 88 Determine whether the function (x) = 2sin x cos x is even, odd, or neither. (Solution on p. 30.) f (x) = 3sin2 x cos x + sec x f (x) = sin x − 2cos2 x Exercise 89 Determine whether the function f is even, odd, or neither. is even, odd, or neither. (Solution on p. 30.) (x) = csc2 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 91 (Solution on p. 30.) sec t csc t 8.6 Real-World Applications Exercise 92 1 600 d , Use the equation to nd The amount of sunlight in a certain city can be modeled by the function h where h represents the hours of sunlight, and d is the day of the year. nd how many hours of sunlight there are on February 10, the 42 = 15cos day of the year. State the period of the function. Exercise 93 represents the hours of sunlight, and d http://https://legacy.cnx.org/content/m49374/1.7/ (Solution on p. 30.) 1 500 d , where h is the day of the year. Use the equation to nd how many The amount of sunlight in a certain city can be modeled by the function h = 16cos OpenStax-CNX module: m49374 27 th hours of sunlight there are on September 24, the 267 day of the year. State the period of the function. Exercise 94 The equation P onds. = 20sin (2πt) + 100 models the blood pressure, P, where (a) Find the blood pressure after 15 seconds. t represents time in sec- (b) What are the maximum and minimum blood pressures? Exercise 95 (Solution on p. 30.) The height of a piston, h, in inches, can be modeled by the equation y the crank angle. Find the height of the piston when the crank = 2cos x+6, where x represents ◦ angle is 55 . Exercise 96 The height of a piston, h,in inches, can be modeled by the equation y = 2cos x+5,where x represents 55 ◦ . the crank angle. Find the height of the piston when the crank angle is http://https://legacy.cnx.org/content/m49374/1.7/ OpenStax-CNX module: m49374 28 Solutions to Exercises in this Module Solution √to Exercise (p. 3) √ 2 2 , cos t sin t = − 2 2 , tan t = = −1, sec t = √ Solution √to Exercise (p. 5) sin π3 = cos π3 = tan π3 = sec π3 csc π3 cot π3 3 2 1 2 √ 3 =2 = = √ 2 3 3 √ 3 3 Solution to Exercise (p. 7) √ √ −7π 4 −7π 4 sin sec = = 2 2 , cos √ 2, csc −7π 4 −7π 4 = = 2 2 , tan √ Solution to Exercise (p. 11) √ − √ 2, csc t = − 2, cot t = −1 2, cot −7π 4 −7π 4 = 1, =1 3 Solution to Exercise (p. 12) −2 Solution to Exercise (p. 13) sin t Solution to Exercise (p. 15) 8 cos t = − 17 , sin t = csc t = 17 15 , cot t = 15 17 , tan t 8 − 15 = − 15 8 Solution to Exercise (p. 17) sin t = −1, cos t = 0, tan t = Undened sec t = Undened,csc t = −1, cot t = 0 Solution √ (p. 18) √ to Exercise sec t = 2, csc t = 2, tan t = 1, cot t = 1 Solution to Exercise (p. 19) ≈ −2.414 Solution to Exercise (p. 20) π 4 and the terminal side of the angle is in quadrants I and III. Thus, π 5π , , the sine and cosine values are equal. 4 4 Yes, when the reference angle is at x = Solution to Exercise (p. 20) Substitute the sine of the angle in for y in the Pythagorean Theorem x negative solution. Solution to Exercise (p. 21) The outputs of tangent and cotangent will repeat every π units. Solution to Exercise (p. 21) √ 2 3 3 Solution to Exercise (p. 21) √ 3 Solution to Exercise (p. 21) √ 2 Solution to Exercise (p. 21) 1 http://https://legacy.cnx.org/content/m49374/1.7/ 2 + y 2 = 1. Solve for x and take the OpenStax-CNX module: m49374 29 Solution to Exercise (p. 21) 2 Solution to Exercise (p. 21) √ 3 3 Solution to Exercise (p. 21) √ −233 Solution to Exercise (p. 21) √ 3 Solution to Exercise (p. 21) √ − 2 Solution to Exercise (p. 22) −1 Solution to Exercise (p. 22) −2 Solution to Exercise (p. 22) √ − 3 3 Solution to Exercise (p. 22) 2 Solution to Exercise (p. 22) √ 3 3 Solution to Exercise (p. 22) −2 Solution to Exercise (p. 22) −1 Solution to√Exercise (p. 22) Ifsin t √ √ = − 2 3 2 , sec t = −3, csc t = − 3 4 2 , tan t = 2 2, cot t = Solution to Exercise (p. 22) √ √ sec t = 2, csc t = 2 3 3 , tan t = Solution to Exercise (p. 22) √ − 3, cot t = √ 2 4 √ 3 3 2 2 Solution to Exercise (p. 22) 3.1 Solution to Exercise (p. 23) 1.4 Solution√ to Exercise (p. 23) √ sin t = 2 2 , cos t = 2 2 , tan t = 1, cot t = 1, sec t = Solution √to Exercise (p. 25) √ sin t = − 3 2 , cos t = − 12 , tan t = Solution to Exercise (p. 25) √ 3, cot t = 0.228 Solution to Exercise (p. 25) 2.414 Solution to Exercise (p. 25) 1.414 Solution to Exercise (p. 26) 1.540 Solution to Exercise (p. 26) 1.556 http://https://legacy.cnx.org/content/m49374/1.7/ √ 2, csc t = 3 3 , sec t √ 2 √ = −2, csc t = − 2 3 3 OpenStax-CNX module: m49374 30 Solution to Exercise (p. 26) sin (t) ≈ 0.79 Solution to Exercise (p. 26) csct ≈ 1.16 Solution to Exercise (p. 26) even Solution to Exercise (p. 26) even Solution to Exercise (p. 26) sin t cos t = tan t Solution to Exercise (p. 26) 13.77 hours, period: 1000π Solution to Exercise (p. 27) 7.73 inches Glossary Denition 1: cosecant the reciprocal of the sine function: on the unit circle, csc t Denition 2: cotangent = y1 , y 6= 0 the reciprocal of the tangent function: on the unit circle, cot t Denition 3: identities = xy , y 6= 0 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 Denition 6: tangent the quotient of the sine and cosine: on the unit circle, tan t http://https://legacy.cnx.org/content/m49374/1.7/ = x1 , x 6= 0 = xy , x 6= 0