Chapter 8 Summary and Review

Key Concepts

1. 1. Identities are useful for changing from one form to another when solving equations, for simplifying expressions, and for finding exact values for trigonometric functions.
   2. 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]

   3. 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]

   4. 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]

   5. 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.
   6. Two functions are called inverse functions if each “undoes” the results of the other function.
   7. 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]
   8. 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]
   9. 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]

   10. Horizontal Line Test.

   11. 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.
   12. A function [latex]f[/latex] has an inverse function if and only if [latex]f[/latex] is one-to-one.

   13. 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]

   14. Inverse Trigonometric Functions.

       [latex]\sin(\sin^{-1}x) = x~~~~\text{for}~ -1 \le x \le 1[/latex]

       [latex]\cos(\cos^{-1}x) = x~~~~\text{for}~ -1 \le x \le 1[/latex]

       [latex]\tan(\tan^{-1}x) = x~~~~\text{for all}~x[/latex]

       However,

       [latex]\sin^{-1}(\sin x)~~\text{may not be equal to}~x[/latex]

       [latex]\cos^{-1}(\cos x)~~\text{may not be equal to}~x[/latex]

       [latex]\tan^{-1}(\tan x)~~\text{may not be equal to}~x[/latex]

   15. The inverse sine function is also called the arcsine function and denoted by [latex]\arcsin (x)\text{.}[/latex] Similarly, the inverse cosine function is sometimes denoted by [latex]\arccos (x)\text{,}[/latex] and the inverse tangent function by [latex]\arctan (x)\text{.}[/latex]
   16. When simplifying expressions involving inverse trigonometric functions, it can often clarify the computations if we assign a name such as [latex]\theta[/latex] or [latex]\phi[/latex] to the inverse trig value.

   17. Three More Functions.

       If [latex]\theta[/latex] is an angle in standard position, and [latex]P(x,y)[/latex] is a point on the terminal side, then we define the following functions.

       [latex]\text{The}~~ \textbf{secant}:~~~~~~ \sec \theta = \dfrac{r}{x}[/latex]

       [latex]\text{The}~~ \textbf{cosecant}:~~~~ \csc \theta = \dfrac{r}{y}[/latex]

       [latex]\text{The}~~ \textbf{cotangent}:~~~ \cot \theta = \dfrac{x}{y}[/latex]

       

   18. Reciprocal Trigonometric Ratios.

       If [latex]\theta[/latex] is one of the acute angles in a right triangle,

       [latex][latex] \begin{aligned}[t] \sec \theta = \dfrac{\text{hypotenuse}}{\text{adjacent}} \\ \csc \theta = \dfrac{\text{hypotenuse}}{\text{opposite}} \\ \cot \theta = \dfrac{\text{adjacent}}{\text{opposite}} \\ \end{aligned}[/latex]

       

   19. Reciprocal Trigonometric Functions.

       [latex]\text{The}~~ \textbf{secant} ~\text{function}:~~~~~~ \sec \theta = \dfrac{1}{\cos \theta}[/latex]

       [latex]\text{The}~~ \textbf{cosecant}~\text{function}:~~~~ \csc \theta = \dfrac{1}{\sin \theta}[/latex]

       [latex]\text{The}~~ \textbf{cotangent}~\text{function}:~~~ \cot \theta = \dfrac{1}{\tan \theta}[/latex]

   20. We can obtain graphs of the secant, cosecant, and cotangent functions as the reciprocals of the three basic functions.
   21. We can solve equations of the form [latex]\sec \theta = k\text{,}[/latex] [latex]\csc \theta = k\text{,}[/latex] and [latex]\cot \theta = k[/latex] by taking the reciprocal of both sides.
   22. If we know one of the trigonometric ratios for an angle, we can use identities to find any of the others.

   23. Cotangent Identity.

       [latex]\cot \theta = \dfrac{1}{\tan \theta} = \dfrac{\cos \theta}{\sin \theta},~~~~\sin \theta \not=0[/latex]

   24. Two More Pythagorean Identities.

       [latex]1 + \tan^2 \theta = \sec^2\theta~~~~~~~~~~1 + \cot^2\theta = \csc^2 \theta[/latex]

       We can often simplify trigonometric expressions by first converting all the trig ratios to sines and cosines.