Stewart Precalc 6e Section 5.5: Inverse Trigonometric Functions and Their Graphs

1 Concepts · Level 1

(a) To define the inverse sine function, we restrict the domain of sine to the interval _______. On this interval the sine function is one-to-one, and its inverse function \(\sin^{-1}\) is defined by \(\sin^{-1} x = y \leq > \sin\) _______ \(=\) _______. For example, \(\sin^{-1}\left(\dfrac{1}{2}\right) = \) _______ because \(\sin\) _______ \(=\) _______.

Answer for 1(a)

(b) To define the inverse cosine function we restrict the domain of cosine to the interval _______. On this interval the cosine function is one-to-one, and its inverse function \(\cos^{-1}\) is defined by \(\cos^{-1} x = y \leq > \cos\) _______ \(=\) _______. For example, \(\cos^{-1}\left(\dfrac{1}{2}\right) = \) _______ because \(\cos\) _______ \(=\) _______.

Answer for 1(b)

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2 Concepts · Level 1

The cancellation property \(\sin^{-1}(\sin x) = x\) is valid for \(x\) in the interval _______. For each of the following, determine if the statement is true.

(a) \(\sin^{-1}(\sin\left(\dfrac{\pi}{3}\right)) = \dfrac{\pi}{3}\)

Answer for 2(a)

(b) \(\sin^{-1}(\sin\left(\dfrac{10 \pi}{3}\right)) = \dfrac{10 \pi}{3}\)

Answer for 2(b)

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3 Skills - Exact Value (Inverse Sine) · Level 2

Find the exact value of each expression, if it is defined.

(a) \(\sin^{-1}(1)\)

Answer for 3(a)

(b) \(\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)\)

Answer for 3(b)

(c) \(\sin^{-1}(2)\)

Answer for 3(c)

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4 Skills - Exact Value (Inverse Sine) · Level 2

Find the exact value of each expression, if it is defined.

(a) \(\sin^{-1}(-1)\)

Answer for 4(a)

(b) \(\sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right)\)

Answer for 4(b)

(c) \(\sin^{-1}(-2)\)

Answer for 4(c)

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5 Skills - Exact Value (Inverse Cosine) · Level 2

Find the exact value of each expression, if it is defined.

(a) \(\cos^{-1}(-1)\)

Answer for 5(a)

(b) \(\cos^{-1}\left(\dfrac{1}{2}\right)\)

Answer for 5(b)

(c) \(\cos^{-1}\left(-\dfrac{\sqrt{3}}{2}\right)\)

Answer for 5(c)

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6 Skills - Exact Value (Inverse Cosine) · Level 2

Find the exact value of each expression, if it is defined.

(a) \(\cos^{-1}\left(\dfrac{\sqrt{2}}{2}\right)\)

Answer for 6(a)

(b) \(\cos^{-1}(1)\)

Answer for 6(b)

(c) \(\cos^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)\)

Answer for 6(c)

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7 Skills - Exact Value (Inverse Tangent) · Level 2

Find the exact value of each expression, if it is defined.

(a) \(\tan^{-1}(-1)\)

Answer for 7(a)

(b) \(\tan^{-1}(\sqrt{3})\)

Answer for 7(b)

(c) \(\tan^{-1}\left(\dfrac{\sqrt{3}}{2}\right)\)

Answer for 7(c)

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8 Skills - Exact Value (Inverse Tangent) · Level 2

Find the exact value of each expression, if it is defined.

(a) \(\tan^{-1}(0)\)

Answer for 8(a)

(b) \(\tan^{-1}(-\sqrt{3})\)

Answer for 8(b)

(c) \(\tan^{-1}\left(-\dfrac{\sqrt{3}}{3}\right)\)

Answer for 8(c)

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9 Skills - Exact Value (Mixed) · Level 2

Find the exact value of each expression, if it is defined.

(a) \(\cos^{-1}\left(-\dfrac{1}{2}\right)\)

Answer for 9(a)

(b) \(\sin^{-1}\left(-\dfrac{\sqrt{2}}{2}\right)\)

Answer for 9(b)

(c) \(\tan^{-1}(1)\)

Answer for 9(c)

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10 Skills - Exact Value (Mixed) · Level 1

Find the exact value of each expression, if it is defined.

(a) \(\cos^{-1}(0)\)

Answer for 10(a)

(b) \(\sin^{-1}(0)\)

Answer for 10(b)

(c) \(\sin^{-1}\left(-\dfrac{1}{2}\right)\)

Answer for 10(c)

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11 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\sin^{-1}\left(\dfrac{2}{3}\right)\) correct to five decimal places, if it is defined.

12 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\sin^{-1}\left(-\dfrac{8}{9}\right)\) correct to five decimal places, if it is defined.

13 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\cos^{-1}\left(-\dfrac{3}{7}\right)\) correct to five decimal places, if it is defined.

14 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\cos^{-1}\left(\dfrac{4}{9}\right)\) correct to five decimal places, if it is defined.

15 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\cos^{-1}(-0.92761)\) correct to five decimal places, if it is defined.

16 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\sin^{-1}(0.13844)\) correct to five decimal places, if it is defined.

17 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\tan^{-1}(10)\) correct to five decimal places, if it is defined.

18 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\tan^{-1}(-26)\) correct to five decimal places, if it is defined.

19 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\tan^{-1}(1.23456)\) correct to five decimal places, if it is defined.

20 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\cos^{-1}(1.23456)\) correct to five decimal places, if it is defined.

21 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\sin^{-1}(-0.25713)\) correct to five decimal places, if it is defined.

22 Skills - Calculator Approximation · Level 2

Use a calculator to find an approximate value of \(\tan^{-1}(-0.25713)\) correct to five decimal places, if it is defined.

23 Skills - Direct Cancellation · Level 1

Evaluate the expression: \(\sin(\sin^{-1}\left(\dfrac{1}{4}\right))\)

24 Skills - Direct Cancellation · Level 1

Evaluate the expression: \(\cos(\cos^{-1}\left(\dfrac{2}{3}\right))\)

25 Skills - Direct Cancellation · Level 1

Evaluate the expression: \(\tan(\tan^{-1}(5))\)

26 Skills - Direct Cancellation · Level 2

Evaluate the expression: \(\sin(\sin^{-1}(5))\)

27 Skills - Direct Cancellation · Level 2

Evaluate the expression: \(\sin(\sin^{-1}\left(\dfrac{3}{2}\right))\)

28 Skills - Direct Cancellation · Level 1

Evaluate the expression: \(\tan(\tan^{-1}\left(\dfrac{3}{2}\right))\)

29 Skills - Inverse Composition · Level 2

Evaluate the expression: \(\cos^{-1}(\cos\left(\dfrac{5 \pi}{6}\right))\)

30 Skills - Inverse Composition · Level 1

Evaluate the expression: \(\tan^{-1}(\tan\left(\dfrac{\pi}{4}\right))\)

31 Skills - Inverse Composition · Level 1

Evaluate the expression: \(\sin^{-1}(\sin\left(-\dfrac{\pi}{6}\right))\)

32 Skills - Inverse Composition · Level 1

Evaluate the expression: \(\tan^{-1}(\tan\left(-\dfrac{\pi}{4}\right))\)

33 Skills - Inverse Composition · Level 3

Evaluate the expression: \(\sin^{-1}(\sin\left(\dfrac{5 \pi}{6}\right))\)

34 Skills - Inverse Composition · Level 2

Evaluate the expression: \(\cos^{-1}(\cos\left(-\dfrac{\pi}{6}\right))\)

35 Skills - Inverse Composition · Level 3

Evaluate the expression: \(\cos^{-1}(\cos\left(\dfrac{17 \pi}{6}\right))\)

36 Skills - Inverse Composition · Level 3

Evaluate the expression: \(\tan^{-1}(\tan\left(\dfrac{4 \pi}{3}\right))\)

37 Skills - Inverse Composition · Level 3

Evaluate the expression: \(\tan^{-1}(\tan\left(\dfrac{2 \pi}{3}\right))\)

38 Skills - Inverse Composition · Level 3

Evaluate the expression: \(\sin^{-1}(\sin\left(\dfrac{11 \pi}{4}\right))\)

39 Skills - Mixed Composition · Level 2

Evaluate the expression: \(\tan(\sin^{-1}\left(\dfrac{1}{2}\right))\)

40 Skills - Mixed Composition · Level 1

Evaluate the expression: \(\cos(\sin^{-1}(0))\)

41 Skills - Mixed Composition · Level 2

Evaluate the expression: \(\cos(\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right))\)

42 Skills - Mixed Composition · Level 2

Evaluate the expression: \(\tan(\sin^{-1}\left(\dfrac{\sqrt{2}}{2}\right))\)

43 Skills - Mixed Composition · Level 2

Evaluate the expression: \(\sin(\tan^{-1}(-1))\)

44 Skills - Mixed Composition · Level 2

Evaluate the expression: \(\sin(\tan^{-1}(-\sqrt{3}))\)

45 Discovery / Discussion / Writing · Level 4

Two Different Compositions. Let \(f\) and \(g\) be the functions \(f(x) = \sin(\sin^{-1} x)\) and \(g(x) = \sin^{-1}(\sin x)\). By the cancellation properties, \(f(x) = x\) and \(g(x) = x\) for suitable values of \(x\). But these functions are not the same for all \(x\). Graph both \(f\) and \(g\) to show how the functions differ. (Think carefully about the domain and range of \(\sin^{-1}\).)

46 Discovery / Discussion / Writing · Level 4

Graphing Inverse Trigonometric Functions. (a) Graph \(y = \sin^{-1} x + \cos^{-1} x\) and make a conjecture. (b) Prove that your conjecture is true.

47 Discovery / Discussion / Writing · Level 4

Graphing Inverse Trigonometric Functions. (a) Graph \(y = \tan^{-1} x + \tan^{-1} \dfrac{1}{x}\) and make a conjecture. (b) Prove that your conjecture is true.

48 Example - Evaluating the Inverse Sine Function · Level 2

Find each value:

(a) \(\sin^{-1}\left(\dfrac{1}{2}\right)\)

Answer for 48(a)

(b) \(\sin^{-1}\left(-\dfrac{1}{2}\right)\)

Answer for 48(b)

(c) \(\sin^{-1}\left(\dfrac{3}{2}\right)\)

Answer for 48(c)

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49 Example - Calculator Approximations of Inverse Sine · Level 2

Find approximate values for (a) \(\sin^{-1}(0.82)\) and (b) \(\sin^{-1}\left(\dfrac{1}{3}\right)\).

50 Example - Evaluating Expressions with Inverse Sine · Level 3

Find each value:

(a) \(\sin^{-1}(\sin\left(\dfrac{\pi}{3}\right))\)

Answer for 50(a)

(b) \(\sin^{-1}(\sin((2 \pi)/3))\)

Answer for 50(b)

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51 Example - Evaluating the Inverse Cosine Function · Level 2

Find each value.

(a) \(\cos^{-1}\left(\dfrac{\sqrt{3}}{2}\right)\)

Answer for 51(a)

(b) \(\cos^{-1}(0)\)

Answer for 51(b)

(c) \(\cos^{-1}\left(\dfrac{5}{7}\right)\)

Answer for 51(c)

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52 Example - Evaluating Expressions with Inverse Cosine · Level 2

Find each value.

(a) \(\cos^{-1}(\cos\left(\dfrac{2 \pi}{3}\right))\)

Answer for 52(a)

(b) \(\cos^{-1}(\cos\left(\dfrac{5 \pi}{3}\right))\)

Answer for 52(b)

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53 Example - Evaluating the Inverse Tangent Function · Level 1

Find each value.

(a) \(\tan^{-1}(1)\)

Answer for 53(a)

(b) \(\tan^{-1}(\sqrt{3})\)

Answer for 53(b)

(c) \(\tan^{-1}(20)\)

Answer for 53(c)

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