Stewart Precalc 6e Section 6.4: Inverse Trigonometric Functions and Right Triangles

1 Concept - Domain and Range of Inverse Trig Functions · Level 1

The inverse sine, inverse cosine, and inverse tangent functions have the following domains and ranges. Fill in the blanks. (a) The function \(\sin^{-1}\) has domain ___ and range ___. (b) The function \(\cos^{-1}\) has domain ___ and range ___. (c) The function \(\tan^{-1}\) has domain ___ and range ___.

2 Concept - Inverse Trig Functions on a Right Triangle · Level 2

In the right triangle shown, the angle \(\theta\) can be written using inverse trigonometric functions. Fill in the ratios using the labeled side lengths in the figure. (a) \(\theta = \sin^{-1}\) (opposite side / hypotenuse) (b) \(\theta = \cos^{-1}\) (adjacent side / hypotenuse) (c) \(\theta = \tan^{-1}\) (opposite side / adjacent side)

3 Skill - Exact Values of Inverse Trig · Level 2

Find the exact value of each expression, if it is defined. (a) \(\sin^{-1}\left(\dfrac{1}{2}\right)\) (b) \(\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)\) (c) \(\tan^{-1}(-1)\)

4 Skill - Exact Values of Inverse Trig · Level 2

Find the exact value of each expression, if it is defined. (a) \(\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)\) (b) \(\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)\) (c) \(\tan^{-1}(-\sqrt{3})\)

5 Skill - Exact Values of Inverse Trig · Level 2

Find the exact value of each expression, if it is defined. (a) \(\sin^{-1}\left(-\dfrac{1}{2}\right)\) (b) \(\cos^{-1}\left(\dfrac{1}{2}\right)\) (c) \(\tan^{-1}\left(\dfrac{\sqrt{3}}{3}\right)\)

6 Skill - Exact Values of Inverse Trig · Level 2

Find the exact value of each expression, if it is defined. (a) \(\sin^{-1}(-1)\) (b) \(\cos^{-1}(1)\)

7 Skill - Approximate Values of Inverse Trig · Level 2

Use a calculator to find an approximate value of \(\cos^{-1}(-0.75)\) rounded to five decimal places, if it is defined.

8 Skill - Approximate Values of Inverse Trig · Level 2

Use a calculator to find an approximate value of \(\cos^{-1}\left(-\dfrac{1}{4}\right)\) rounded to five decimal places, if it is defined.

9 Skill - Approximate Values of Inverse Trig · Level 2

Use a calculator to find an approximate value of \(\sin^{-1}\left(\dfrac{1}{3}\right)\) rounded to five decimal places, if it is defined.

10 Skill - Approximate Values of Inverse Trig · Level 2

Use a calculator to find an approximate value of \(\tan^{-1}(3)\) rounded to five decimal places, if it is defined.

11 Skill - Approximate Values of Inverse Trig · Level 2

Use a calculator to find an approximate value of \(\tan^{-1}(-4)\) rounded to five decimal places, if it is defined.

12 Skill - Approximate Values of Inverse Trig · Level 2

Use a calculator to find an approximate value of \(\cos^{-1}(3)\) rounded to five decimal places, if it is defined.

13 Skill - Approximate Values of Inverse Trig · Level 2

Use a calculator to find an approximate value of \(\sin^{-1}(-2)\) rounded to five decimal places, if it is defined.

14 Skill - Angle from Right Triangle · Level 2

Find the angle \(\theta\) in degrees, rounded to one decimal place, using the side lengths shown in the right triangle.

15 Skill - Angle from Right Triangle · Level 2

Find the angle \(\theta\) in degrees, rounded to one decimal place, using the side lengths shown in the right triangle.

16 Skill - Angle from Right Triangle · Level 2

Find the angle \(\theta\) in degrees, rounded to one decimal place, using the side lengths shown in the right triangle.

17 Skill - Solving Equations on an Interval · Level 3

Find all angles \(\theta\) between \(0^{\circ}\) and \(180^{\circ}\) satisfying \(\sin \theta = \dfrac{1}{2}\).

18 Skill - Solving Equations on an Interval · Level 3

Find all angles \(\theta\) between \(0^{\circ}\) and \(180^{\circ}\) satisfying \(\sin \theta = \dfrac{\sqrt{3}}{2}\).

19 Find the angle theta · Level 2

Find the angle \(\theta\) in degrees (rounded to one decimal place), where \(0^{\circ} \leq \theta \leq 90^{\circ}\), such that \(\sin \theta = \dfrac{1}{4}\).

20 Find the angle theta · Level 2

Find the angle \(\theta\) in degrees (rounded to one decimal place), where \(0^{\circ} \leq \theta \leq 90^{\circ}\), such that \(\cos \theta = 0.7\).

21 Find the angle theta · Level 2

Find the angle \(\theta\) in degrees (rounded to one decimal place), where \(0^{\circ} \leq \theta \leq 90^{\circ}\), such that \(\cos \theta = \dfrac{1}{9}\).

22 Exact value of inverse trig expression · Level 2

Find the exact value of the expression \(\sin(\cos^{-1}\left(\dfrac{3}{5}\right))\).

23 Exact value of inverse trig expression · Level 2

Find the exact value of the expression \(\tan(\sin^{-1}\left(\dfrac{4}{5}\right))\).

24 Exact value of inverse trig expression · Level 2

Find the exact value of the expression \(\sec(\sin^{-1}\left(\dfrac{12}{13}\right))\).

25 Exact value of inverse trig expression · Level 2

Find the exact value of the expression \(\csc(\cos^{-1}\left(\dfrac{7}{25}\right))\).

26 Exact value of inverse trig expression · Level 2

Find the exact value of the expression \(\tan(\sin^{-1}\left(\dfrac{12}{13}\right))\).

27 Exact value of inverse trig expression · Level 2

Find the exact value of the expression \(\cot(\sin^{-1}\left(\dfrac{2}{3}\right))\).

28 Rewrite as algebraic expression · Level 3

Rewrite the expression \(\cos(\sin^{-1}(x))\) as an algebraic expression in \(x\).

29 Rewrite as algebraic expression · Level 3

Rewrite the expression \(\sin(\tan^{-1}(x))\) as an algebraic expression in \(x\).

30 Rewrite as algebraic expression · Level 3

Rewrite the expression \(\tan(\sin^{-1}(x))\) as an algebraic expression in \(x\).

31 Rewrite as algebraic expression · Level 3

Rewrite the expression \(\cos(\tan^{-1}(x))\) as an algebraic expression in \(x\).

32 Application - Leaning Ladder · Level 3

Leaning Ladder. A 20-ft ladder is leaning against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle of elevation of the ladder? How high does the ladder reach on the building?

33 Application - Angle of Sun · Level 2

Angle of the Sun. A 96-ft tree casts a shadow that is 120 ft long. What is the angle of elevation of the sun?

34 Application - Space Shuttle · Level 3

Height of the Space Shuttle. An observer views the space shuttle from a distance of 2 mi from the launch pad.

(a) Express the height of the space shuttle as a function of the angle of elevation \(\theta\).

Answer for 34(a)

(b) Express the angle of elevation \(\theta\) as a function of the height \(h\) of the space shuttle.

Answer for 34(b)

Enter your answer directly below each part above.

35 Application - Pole Shadow · Level 3

Height of a Pole. A 50-ft pole casts a shadow as shown in the figure.

(a) Express the angle of elevation \(\theta\) of the sun as a function of the length \(s\) of the shadow.

Answer for 35(a)

(b) Find the angle \(\theta\) of elevation of the sun when the shadow is 20 ft long.

Answer for 35(b)

Enter your answer directly below each part above.

36 Application - Hot-Air Balloon · Level 3

Height of a Balloon. A 680-ft rope anchors a hot-air balloon as shown in the figure.

(a) Express the angle \(\theta\) as a function of the height \(h\) of the balloon.

Answer for 36(a)

(b) Find the angle \(\theta\) if the balloon is 500 ft high.

Answer for 36(b)

Enter your answer directly below each part above.

37 Application - Satellite View · Level 4

View from a Satellite. The figures indicate that the higher the orbit of a satellite, the more of the earth the satellite can "see." Let \(\theta\), \(s\), and \(h\) be as in the figure, and assume the earth is a sphere of radius 3960 mi.

(a) Express the angle \(\theta\) as a function of \(h\).

Answer for 37(a)

(b) Express the distance \(s\) as a function of \(\theta\).

Answer for 37(b)

(c) Express the distance \(s\) as a function of \(h\). [Hint: Find the composition of the functions in parts (a) and (b).]

Answer for 37(c)

(d) If the satellite is 100 mi above the earth, what is the distance \(s\) that it can see?

Answer for 37(d)

(e) How high does the satellite have to be to see both Los Angeles and New York, 2450 mi apart?

Answer for 37(e)

Enter your answer directly below each part above.

38 Application - Surfing Wave · Level 4

Surfing the Perfect Wave. For a wave to be surfable, it can't break all at once. Robert Guza and Tony Bowen have shown that a wave has a surfable shoulder if it hits the shoreline at an angle \(\theta\) given by \(\theta = \sin^{-1}(\dfrac{1}{(2 n + 1) \tan \beta})\) where \(\beta\) is the angle at which the beach slopes down and where \(n = 0, 1, 2, ...\)

(a) For \(\beta = 10^{\circ}\), find \(\theta\) when \(n = 3\).

Answer for 38(a)

(b) For \(\beta = 15^{\circ}\), find \(\theta\) when \(n = 2\), \(3\), and \(4\). Explain why the formula does not give a value for \(\theta\) when \(n = 0\) or \(1\).

Answer for 38(b)

Enter your answer directly below each part above.

39 Discovery - Inverse Trig on Calculator · Level 3

Inverse Trigonometric Functions on a Calculator. Most calculators do not have keys for \(\sec^{-1}\), \(\csc^{-1}\), or \(\cot^{-1}\). Prove the following identities, and then use these identities and a calculator to find \(\sec^{-1}(2)\), \(\csc^{-1}(3)\), and \(\cot^{-1}(4)\). \(\sec^{-1}(x) = \cos^{-1}\left(\dfrac{1}{x}\right), \quad x \geq 1\) \(\csc^{-1}(x) = \sin^{-1}\left(\dfrac{1}{x}\right), \quad x \geq 1\) \(\cot^{-1}(x) = \tan^{-1}\left(\dfrac{1}{x}\right), \quad x > 0\)

40 Example - Evaluating Inverse Trigonometric Functions · Level 2

Find the exact value of each expression. (a) \(\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)\) (b) \(\cos^{-1}\left(-\dfrac{1}{2}\right)\) (c) \(\tan^{-1} 1\)

41 Example - Evaluating Inverse Trigonometric Functions · Level 2

Find approximate values for the given expression. (a) \(\sin^{-1}(0.71)\) (b) \(\tan^{-1}(2)\) (c) \(\cos^{-1}(2)\)

42 Example - Finding an Angle in a Right Triangle · Level 2

Find the angle \(\theta\) in the right triangle shown in Figure 2 (the side opposite \(\theta\) has length 10 and the hypotenuse has length 50).

43 Example - Solving for an Angle in a Right Triangle · Level 2

A 40-ft ladder leans against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle formed by the ladder and the building?

44 Example - The Angle of a Beam of Light · Level 3

A lighthouse is located on an island that is 2 mi off a straight shoreline (see Figure 4). Express the angle \(\theta\) formed by the beam of light and the shoreline in terms of the distance \(d\) in the figure.

45 Example - Solving a Basic Trigonometric Equation on an Interval · Level 3

Find all angles \(\theta\) between \(0^{\circ}\) and \(180^{\circ}\) satisfying the given equation. (a) \(\sin \theta = 0.4\) (b) \(\cos \theta = 0.4\)

46 Example - Composing Trigonometric Functions and Their Inverses · Level 3

Find \(\cos(\sin^{-1}\left(\dfrac{3}{5}\right))\).

47 Example - Composing Trigonometric Functions and Their Inverses · Level 3

Write \(\sin(\cos^{-1} x)\) and \(\tan(\cos^{-1} x)\) as algebraic expressions in \(x\) for \(-1 \leq x \leq 1\).

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