2. Trigonometric Functions on Right Triangles

CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. II. 1. A. B. C. 2. S 70° W 3. 4. D. E. F. 5. 6. 7. G. H. 8. 9. 10. I. J.

CONCEPT PREVIEW  Fill in the blank(s) to correctly complete each sentence. Given tan θ = 1/cot θ , two equivalent forms of this identity are cot θ = 1/______ and tan θ . ______ = 1 .

CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. II. 1. A. B. C. 2. 3. 4. D. E. F. 5. N 70° W 6. 7. G. H. 8. 9. 10. I. J.

CONCEPT PREVIEW  Determine whether each statement is possible or impossible. sin θ = 1/2 , csc θ = 2

CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. II. 1. A. B. C. 2. 3. 4. D. E. F. 5. 6. 7. G. H. 8. 270° 9. 10. I. J.

CONCEPT PREVIEW  Determine whether each statement is possible or impossible. cos θ = 1.5

CONCEPT PREVIEW Match the measure of bearing in Column I with the appropriate graph in Column II. I. II. 1. A. B. C. 2. 3. 4. D. E. F. 5. 6. 7. G. H. 8. 9. 10. N 70° E I. J.

CONCEPT PREVIEW  Determine whether each statement is possible or impossible. sin² θ + cos² θ = 2

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (-4, 0)

Use the figure shown to solve Exercises 13–16. Find the bearing from O to A.

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. csc θ , given that sin θ = ―3/7

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (0, -2)

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (-3, -3)

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. cot θ , given that tan θ = 18

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (2, 2)

Solve each problem. See Examples 1 and 2. Distance Traveled by a Ship A ship travels 55 km on a bearing of 27° and then travels on a bearing of 117° for 140 km. Find the distance from the starting point to the ending point.

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. sin θ , given that csc θ = √24/3

Solve each problem. See Examples 1 and 2. Distance between Two Ships Two ships leave a port at the same time. The first ship sails on a bearing of 52° at 17 knots and the second on a bearing of 322° at 22 knots. How far apart are they after 2.5 hr?

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. sin θ , given that csc θ = 1.25

Solve each problem. See Examples 1 and 2. Distance between Two Ships A ship leaves its home port and sails on a bearing of S 61°50'. Another ship leaves the same port at the same time and sails on a bearing of N 28°10'E. If the first ship sails at 24.0 mph and the second sails at 28.0 mph, find the distance between the two ships after 4 hr.

Concept Check What is wrong with the following item that appears on a trigonometry test? "Find sec θ , given that cos θ = 3/2 . "

Solve each problem. See Examples 1 and 2. Flying Distance The bearing from A to C is N 64° W. The bearing from A to B is S 82° W. The bearing from B to C is N 26° E. A plane flying at 350 mph takes 1.8 hr to go from A to B. Find the distance from B to C.

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2. 84°

Solve each problem. See Examples 1 and 2. Distance between Two Cities The bearing from Atlanta to Macon is S 27° E, and the bearing from Macon to Augusta is N 63° E. An automobile traveling at 62 mph needs 1¼ hr to go from Atlanta to Macon and 1¾ hr to go from Macon to Augusta. Find the distance from Atlanta to Augusta.

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2. 178°

Solve each problem. See Examples 3 and 4. Height of an Antenna A scanner antenna is on top of the center of a house. The angle of elevation from a point 28.0 m from the center of the ho​​use to the top of the antenna is 27°10', and the angle of elevation to the bottom of the antenna is 18°10'. Find the height of the antenna.

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2. ―15°

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2. ―345°

Solve each problem. See Examples 3 and 4. Distance through a Tunnel A tunne​​l is to be built from point A to point B. Both A and B are visible from C. If AC is 1.4923 mi and BC is 1.0837 mi, and if C is 90°, find the measures of angles A and B.

Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable. cos θ = ―5/8 , and θ is in quadrant III

Identify the quadrant (or possible quadrants) of an angle θ  that satisfies the given conditions. See Example 3. cos θ > 0 , sec θ > 0

Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable. sec θ = ―√5 , and θ is in quadrant II

Give all six trigonometric function values for each angle θ . Rationalize denominators when applicable. sec θ = 5/4 , and θ is in quadrant IV

Identify the quadrant (or possible quadrants) of an angle θ  that satisfies the given conditions. See Example 3. tan θ < 0 , cos θ < 0

Determine whether each statement is possible or impossible. a. sec θ = ―2/3

Determine whether each statement is possible or impossible. b. tan θ = 1.4

Determine whether each statement is possible or impossible. c. cos θ = 5

Solve each problem. See Examples 3 and 4. The figure to the right indicates that the equation of a line passing through the point (a, 0) and making an angle θ with the x-axis is y = (tan θ) (x - a). Find an eq​​uation of the line passing through the point (5, 0) that makes an angle of 15° with the x-axis.

Identify the quadrant (or possible quadrants) of an angle θ  that satisfies the given conditions. See Example 3. csc θ > 0 , cot θ > 0

Identify the quadrant (or possible quadrants) of an angle θ  that satisfies the given conditions. See Example 3. tan θ < 0 , cot θ < 0

Solve each problem. Height of a Lunar Peak The lunar mountain peak Huygens has a height of 21,000 ft. The shadow of Huygens on a photograph was 2.8 mm, while the nearby mountain Bradley had a shadow of 1.8 mm on the same photograph. Calculate the height of Bradley. (Data from Webb, T., Celestial Objects for Common Telescopes, Dover Publications.)

Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.) Length of Sides of an Isosceles Triangle An isosceles triangle has a base of length 49.28 m. The angle opposite the base is 58.746°. Find the length of each of the two equal sides.

Determine whether each statement is possible or impossible. See Example 4. sin θ = 3

Determine whether each statement is possible or impossible. See Example 4. tan θ = 0.93

Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.) Find a formula for h in terms of k, A, and B. Assume A < B.

Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.) Create a right triangle problem whose solution can be found by evaluating θ if sin θ = ¾.

In Exercises 61–62, use the figures shown to find the bearing from O to A.

Determine whether each statement is possible or impossible. See Example 4. csc θ = 100

Determine whether each statement is possible or impossible. See Example 4. cot θ = ―6

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find csc θ , given that cot θ = ―1/2 and θ is in quadrant IV.

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find cot θ , given that csc θ = ―1.45 and θ is in quadrant III.

If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. tan θ = ―15/8 , and θ is in quadrant II .

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. sin θ = √5/7 , and θ is in quadrant I .

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. sin θ = √2/6 , and cos θ < 0

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. csc θ = ―3 , and cos θ > 0

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. cos θ = 1

Concept Check Suppose that 90° < θ < 180° .   Find the sign of each function value. tan θ/2

Concept Check Suppose that 90° < θ < 180° .   Find the sign of each function value. cot (θ + 180°)

Concept Check Suppose that 90° < θ < 180° .   Find the sign of each function value. cos ( ―θ)

Concept Check Suppose that ―90° < θ < 90° .   Find the sign of each function value. sec θ/2

Concept Check Suppose that ―90° < θ < 90° .   Find the sign of each function value. sec(―θ)

Concept Check Suppose that ―90° < θ < 90° .   Find the sign of each function value. cos(θ―180°)

Concept Check Find a solution for each equation. tan (3θ ― 4°) = 1 / [cot(5θ ― 8°)]

Concept Check Find a solution for each equation. sin(4θ + 2°) csc(3θ + 5°) = 1

Concept Check Find a solution for each equation. sec(2θ + 6°) cos(5θ + 3°) = 1

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

In Exercises 1–8, use the Pythagorean Theorem to find the length of the missing side of each right triangle. Then find the value of each of the six trigonometric functions of θ.

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

cos 30°

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

tan 30°

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

cot 𝜋/3

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

sin 𝜋/4 - cos 𝜋/4

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

tan 𝜋/4 + csc 𝜋/6

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. tan 𝜋/3 - ___1___ 2 sec 𝜋/6

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. 1 + sin² 40° + sin² 50°

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. csc 37° sec 53° - tan 53° cot 37°

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. cos 12° sin 78° + cos 78° sin 12°

In Exercises 69–70, express the exact value of each function as a single fraction. Do not use a calculator. If f(θ) = 2 cos θ - cos 2θ, find f(𝜋/6).

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (-4, 3)

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (3, 7)

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (5, -5)

In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ. (-1, -3)

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. csc 𝜋

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. cos 3𝜋 2

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. cot 𝜋 2

In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. tan 𝜋 2

In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies. sin θ > 0, cos θ > 0

In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies. sin θ < 0, cos θ < 0

In Exercises 17–22, let θ be an angle in standard position. Name the quadrant in which θ lies. tan θ < 0, cos θ < 0

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. cos θ = -3/5, θ in quadrant III

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. sin θ = 5/13, θ in quadrant II

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. cos θ = 8/17, 270° < θ < 360°

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = -2/3, sin θ > 0

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = 4/3, cos θ < 0

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. sec θ = -3, tan θ > 0

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. tan(-t) - tan t

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. 3 cos(-t) - cos t

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. sin(-t - 2𝜋) - cos(-t - 4𝜋) - tan(-t - 𝜋)

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)

Use the triangle to find each of the six trigonometric functions of θ.

In Exercises 23–26, find the exact value of each expression. Do not use a calculator. cos² 𝜋 - tan² 𝜋 4 4

In Exercises 23–26, find the exact value of each expression. Do not use a calculator. cos 2𝜋 sec 2𝜋 9 9

In Exercises 37–38, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ, or state that the function is undefined. (0, -1)

If sin θ = a and cos θ = b, represent each of the following in terms of a and b. tan θ - sec θ

CONCEPT PREVIEW The terminal side of an angle θ in standard position passes through the point (― 3,― I3) Use the figure to find the following values. Rationalize denominators when applicable. r

CONCEPT PREVIEW The terminal side of an angle θ in standard position passes through the point (― 3,― I3) Use the figure to find the following values. Rationalize denominators when applicable. tan θ

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―12 , ―5)

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―8 , 15)

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (0, ―3)

Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) III , y/r

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) IV , x/y

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) IV , x/r

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) II , y/x

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) III , r/y

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) I , y/r

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) I , r/y

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. 2x + y = 0 , x ≥ 0

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. ―5x ― 3y = 0 , x ≤ 0

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . See Example 3. x = 0 , y ≥ 0

Find the indicated function value. If it is undefined, say so. See Example 4. sin 90°

Find the indicated function value. If it is undefined, say so. See Example 4. sec 180°

Find the indicated function value. If it is undefined, say so. See Example 4. sin(―270°)

Find the indicated function value. If it is undefined, say so. See Example 4. tan 450°

Find the indicated function value. If it is undefined, say so. See Example 4. cos 1800°

Find the indicated function value. If it is undefined, say so. See Example 4. sec 1800°

Use trigonometric function values of quadrantal angles to evaluate each expression. 3 sec 180° ― 5 tan 360°

Use trigonometric function values of quadrantal angles to evaluate each expression. tan 360° + 4 sin 180° + 5(cos 180°)²

Use trigonometric function values of quadrantal angles to evaluate each expression. (sin 180°)² + (cos 180°)²

Use trigonometric function values of quadrantal angles to evaluate each expression. (sec 180°)² ― 3 (sin 360°)² + cos 180°

Use trigonometric function values of quadrantal angles to evaluate each expression. ―3(sin 90°)⁴ + 4(cos 180°)³

Use trigonometric function values of quadrantal angles to evaluate each expression. [cos(―180°)]² + [sin(― 180°)]²

If n is an integer, n • 180° represents an integer multiple of 180°, (2n + 1) • 90° represents an odd integer multiple of 90° , and so on. Determine whether each expression is equal to 0, 1, or ―1, or is undefined. sin[n • 180°]

If n is an integer, n • 180° represents an integer multiple of 180°, (2n + 1) • 90° represents an odd integer multiple of 90° , and so on. Determine whether each expression is equal to 0, 1, or ―1, or is undefined. sin[270° + n • 360°]

If n is an integer, n • 180° represents an integer multiple of 180°, (2n + 1) • 90° represents an odd integer multiple of 90° , and so on. Determine whether each expression is equal to 0, 1, or ―1, or is undefined. cot[n • 180°]

If n is an integer, n • 180° represents an integer multiple of 180°, (2n + 1) • 90° represents an odd integer multiple of 90° , and so on. Determine whether each expression is equal to 0, 1, or ―1, or is undefined. cos[n • 360°]

Find the six trigonometric function values for each angle. Rationalize denominators when applicable.

Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (3 , ―4)

Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (―8 , 15)

Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (6√3 , ―6)

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . 12x + 5y = 0 , x ≥ 0 .