7.4 Inverse Trigonometric Functions

7.4.1 Learning Objectives

By the end of this section, you should be able to:

  • state the restricted domains and ranges of inverse trigonometric functions;
  • evaluate inverse sine, cosine, and tangent exactly;
  • evaluate compositions involving trigonometric and inverse functions;
  • simplify expressions by constructing triangles; and
  • use inverse functions to solve applied problems.

7.4.2 Principal-Value Ranges

Trigonometric functions must be restricted to become one-to-one:

\[ y=\sin^{-1}x\quad\Longleftrightarrow\quad \sin y=x,\qquad -\frac{\pi}{2}\le y\le\frac{\pi}{2}, \]

\[ y=\cos^{-1}x\quad\Longleftrightarrow\quad \cos y=x,\qquad 0\le y\le\pi, \]

\[ y=\tan^{-1}x\quad\Longleftrightarrow\quad \tan y=x,\qquad -\frac{\pi}{2}<y<\frac{\pi}{2}. \]

The notation \(\sin^{-1}x\) means inverse sine, not \(1/\sin x\).

Example 7.10 Evaluating Principal Values

Evaluate \(\sin^{-1}(-1/2)\), \(\cos^{-1}(-1/2)\), and \(\tan^{-1}(1)\).

Solution

\[ \boxed{\sin^{-1}(-1/2)=-\pi/6,\quad \cos^{-1}(-1/2)=2\pi/3,\quad \tan^{-1}(1)=\pi/4} \]

Exercise 7.10 Evaluate \(\cos^{-1}(\sqrt2/2)\).

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Answer: \(\boxed{\pi/4}\)

7.4.3 Compositions of Functions

On the restricted domains,

\[ \sin(\sin^{-1}x)=x,\quad \cos(\cos^{-1}x)=x,\quad \tan(\tan^{-1}x)=x. \]

The reverse compositions require attention to principal ranges. For example, \(\sin^{-1}(\sin x)\) equals \(x\) only when \(x\in[-\pi/2,\pi/2]\).

Example 7.11 Evaluating a Reverse Composition

Evaluate \(\sin^{-1}(\sin(5\pi/6))\).

Solution

\[ \sin(5\pi/6)=1/2. \]

The angle in the inverse-sine range with sine \(1/2\) is \(\pi/6\):

\[ \boxed{\pi/6}. \]

Exercise 7.11 Evaluate \(\cos^{-1}(\cos(4\pi/3))\).

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Answer: Since \(\cos(4\pi/3)=-1/2\), the principal value is \(\boxed{2\pi/3}\).

7.4.4 Algebraic Inverse-Trigonometric Expressions

To simplify an expression such as \(\cos(\sin^{-1}x)\), let \(\theta=\sin^{-1}x\), build a right triangle, and use the principal range to determine signs.

Example 7.12 Using a Reference Triangle

Simplify \(\cos(\tan^{-1}x)\).

Solution

Let \(\theta=\tan^{-1}x\), so \(\tan\theta=x/1\). A reference triangle has opposite side \(x\), adjacent side \(1\), and hypotenuse \(\sqrt{1+x^2}\). Cosine is positive on the inverse-tangent range:

\[ \boxed{\cos(\tan^{-1}x)=\frac1{\sqrt{1+x^2}}}. \]

Exercise 7.12 Simplify \(\tan(\sin^{-1}x)\).

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Answer: \(\boxed{\dfrac{x}{\sqrt{1-x^2}}}\), for \(-1<x<1\).

7.4.5 Conceptual Takeaways

  • Restricted domains make inverse trigonometric functions possible.
  • Each inverse function returns a principal angle.
  • Direct inverse compositions cancel on their domains.
  • Reverse compositions may return a coterminal or related principal angle.
  • Reference triangles convert inverse-trigonometric expressions to algebraic forms.

7.4.6 Skills You Should Be Able to Do

  • State inverse-function domains and ranges.
  • Evaluate exact principal values.
  • Calculate direct and reverse compositions.
  • Simplify algebraic inverse-trigonometric expressions.
  • Use calculator approximations in degree or radian mode.

7.4.7 Practice Problems with Solutions

  1. Evaluate \(\sin^{-1}(\sqrt3/2)\).

    Show Solution

    The angle in \([-\pi/2,\pi/2]\) is \(\boxed{\pi/3}\).

  2. Evaluate \(\cos^{-1}(0)\).

    Show Solution

    The angle in \([0,\pi]\) with cosine \(0\) is \(\boxed{\pi/2}\).

  3. Evaluate \(\tan^{-1}(-\sqrt3)\).

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    The principal angle is \(\boxed{-\pi/3}\).

  4. Evaluate \(\cos(\cos^{-1}(-0.4))\).

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    Direct composition gives \(\boxed{-0.4}\).

  5. Evaluate \(\tan^{-1}(\tan(3\pi/4))\).

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    \(\tan(3\pi/4)=-1\). The principal angle with tangent \(-1\) is

    \[ \boxed{-\pi/4}. \]

  6. Simplify \(\sin(\cos^{-1}x)\).

    Show Solution

    Let \(\theta=\cos^{-1}x\). Then \(\cos\theta=x\), and sine is non-negative on \([0,\pi]\):

    \[ \boxed{\sin(\cos^{-1}x)=\sqrt{1-x^2}}. \]