Chapter 8: More Functions and Identities
Chapter 8 Summary and Review
Key Concepts
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- Identities are useful for changing from one form to another when solving equations, for simplifying expressions, and for finding exact values for trigonometric functions.
- it is not true in general that [latex]\cos (\alpha + \beta)[/latex] is equal to [latex]\cos \alpha + \cos \beta[/latex] for all angles [latex]\alpha[/latex] and [latex]\beta{,}[/latex] or that [latex]\sin (\alpha + \beta)[/latex] is equal to [latex]\sin \alpha + \sin \beta\text{.}[/latex]
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Sum and Difference of Angles Identities.
[latex]\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta[/latex]
[latex]\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta[/latex]
[latex]\tan (\alpha + \beta) = \dfrac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}[/latex]
[latex]\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta[/latex]
[latex]\sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta[/latex]
[latex]\tan (\alpha - \beta) = \dfrac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}[/latex]
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Double Angle Identities.
[latex]\cos 2\theta = \cos^2 \theta - \sin^2 \theta \sin 2\theta = 2\sin \theta \cos \theta\\ = 2\cos^2 \theta -1 \tan 2\theta = \dfrac {2\tan \theta}{1-\tan^2 \theta}\\ = 1-2\sin^2 \theta[/latex]
- Using one of the calculator keys [latex]SIN^{-1},~ COS^{-1}\text{,}[/latex] or [latex]TAN^{-1}[/latex] performs the inverse operation for computing a sine, cosine, or tangent.
- Two functions are called inverse functions if each “undoes” the results of the other function.
- If [latex]y=f(x)[/latex] is a function, we can often find a formula for the inverse function by interchanging [latex]x[/latex] and [latex]y[/latex] in the formula for the function and then solving for [latex]y\text{.}[/latex]
- The graphs of [latex]y=f(x)[/latex] and [latex]y=f^{-1}(x)[/latex] are symmetric about the line [latex]y=x\text{.}[/latex]
- The domain of [latex]f^{-1}[/latex] is the same as the range of [latex]f\text{,}[/latex] and the range of [latex]f^{-1}[/latex] is the same as the domain of [latex]f\text{.}[/latex]
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Horizontal Line Test.
- A function passes the Horizontal Line Test if every horizontal line intersects the graph at most once. In that case, there is only one [latex]x[/latex]-value for each [latex]y[/latex]-value, and the function is called one-to-one.
- A function [latex]f[/latex] has an inverse function if and only if [latex]f[/latex] is one-to-one.
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Definitions of the Inverse Trig Functions.
[latex]\sin^{-1}x=\theta~~~\text{if and only if}~~~~\sin \theta = x~~~\text{and}~~~\dfrac{-\pi}{2} \le \theta \le \dfrac{\pi}{2}[/latex]
[latex]\cos^{-1}x=\theta~~~\text{if and only if}~~~~\cos \theta = x~~~\text{and}~~~0 \le \theta \le \pi[/latex]
[latex]\tan^{-1}x=\theta~~~\text{if and only if}~~~~\tan \theta = x~~~\text{and}~~~\dfrac{-\pi}{2} \lt \theta \lt \dfrac{\pi}{2}[/latex]
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Inverse Trigonometric Functions.