LESSON 1 • Trigonometric Functions 471

Check Your UnderstandingCheck Your UnderstandingTerri is flying a kite and has let out 500 feet of string. Her end of the string is 3 feet off the ground.

a. If ∠KIT has a measure of 40°, approximately how high off the ground is the kite?

b. As the wind picks up, Terri is able to fly the kite at a 56° angle with the horizontal. Approximately how high is the kite?

c. What is the highest Terri could fly the kite on 500 feet of string? What would be the measure of ∠KIT then?

d. Your answers in Parts a–c are estimates. When you actually fly a kite, what are some factors that might cause your answers to be somewhat inaccurate?

IInvest invest iggationation 33 What’s the Angle? What’s the Angle?In Investigation 2, you used trigonometric functions to determine an unknown or inaccessible distance in situations that could be modeled with right triangles. In such situations, trigonometric functions can also be used to determine the measure of an angle when you know the lengths of two sides of a triangle. As you work on this investigation, look for answers to the following question:

How can trigonometric functions be used to determine the measure of an acute angle in a right triangle

when the lengths of two sides are known?

1 In Investigation 2, you explored ways to use trigonometric functions to indirectly measure the height of Chicago’s Bat Column sculpture. Cal read some literature that said the Bat Column’s height BD is about 30 meters.

a. Cal sighted to the top of the sculpture from a point A, 25 meters from point C and 1.5 meters above the ground. Experiment with your calculator to estimate, to the nearest degree, the measure of the angle of elevation that Cal should expect at point A.

b. Suppose Cal sighted from a point 18 meters from point C and 1.5 meters above the ground. Use the same method as in Part a to estimate the measure of the angle of elevation from this point.

c. Check your answers in Parts a and b using the table on page 465.

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Name: _______________________________

Unit 6Lesson 1

Investigation 3

472 UNIT 7 • Trigonometric Methods

2 Using your calculator, estimate (to the nearest degree) the measure of acute ∠B for each of the following trigonometric functions. Check your estimate in each case using the table on page 465.

a. sin B = 3 _ 4 b. cos B = 1 _ 2 c. tan B = 2

3 You can also use the inverse function capabilities of your calculator to produce the acute angle measure when you know a trigonometric function value as in Problem 2. The function sin-1 (read “inverse sine”) is related to the sine function for acute angles in a way similar to the way the square root and squaring operation are related. One function “undoes” the other. Thus, sin-1 x is the acute angle whose sine is x. The inverse cosine (denoted cos-1) and inverse tangent (denoted tan-1) are similarly related to the cosine and tangent functions.

a. Suppose you know sin A = 4 _ 5 . Use the sin-1 function of your calculator to compute the measure of ∠A. (Make certain your

calculator is set in degree mode.)

b. Use inverse trigonometric functions to find the measure of ∠B that corresponds to each of the function values given in Problem 2. Compare these values to the values you obtained in that problem.

c. Use the calculator display at the right to find sin 37.58950296°.

d. Use your calculator to find the degree measure of the acute angle in each of the following cases, where possible.

i. tan B = 1.84

ii. sin A = 0.852

iii. sin A = 2.15

iv. cos B = 0.213

4 House painters make decisions not only about the type of paint to use but also about what equipment to use. Suppose a house painter needs to buy a new extension ladder that will reach up to a vertical height of 30 feet when leaned against the side of a house. For safety reasons, it is recommended that the ladder is placed so the angle it makes with level ground measures 75°.

a. What is the minimum fully-extended length of a ladder that meets these requirements?

b. Suppose the painter’s assistant extends the new ladder to 28 feet and leans it against a house from a point 11.5 feet from the house.

i. What angle does the ladder make with the ground?

ii. At what vertical height from the ground does the ladder meet the side of the house?

iii. To what length should the ladder be adjusted if it is to reach the same point on the side of the house but make the recommended 75° angle with the ground?

iv. How far will the ladder be placed from the house after it is adjusted as in part iii?

LESSON 1 • Trigonometric Functions 473

Check Your UnderstandingCheck Your UnderstandingA person on an oil-drilling ship in the Gulf of Mexico sees a semi-submersible platform with a tower on top of it. The tower stands 130 meters above the platform floor.

a. If the observer’s position on the boat is 15 meters under the floor of the platform and the angle of elevation to the top of the rig is 21°, what three distances can you find? Find them.

b. Suppose the boat moves so the observer is 200 meters from the center line of the tower. What is the angle of elevation now?

Summarize the Mathematics

The sine, cosine, and tangent functions are useful in finding the measures of acute angles of a right triangle using the lengths of two sides. Refer to the right triangle below to summarize your thinking about such situations.

B

A Cb

ac

a If you knew a and c, how would you use that information to find the measure of ∠B? How could you then find m∠A? How could you find b?

b If you knew b and c, how would you use that information to find the measure of ∠B? How could you then find m∠A?

c If you knew a and b, how would you use that information to find the measure of ∠A?

Be prepared to explain your methods to the entire class.

Platform

21°