Chapter 5 Summary and Review

Key Concepts

1. Expressions containing trig ratios can be simplified or evaluated like other algebraic expressions. To simplify an expression containing trig ratios, we treat each ratio as a single variable.
2. The parentheses in an expression such as $\sin (X+Y)$ indicate function notation, not multiplication.
3. We write ${\cos }^2\theta$ to denote $(\cos \theta {)}^2,$ and ${\cos }^n\theta$ to denote $(\cos \theta {)}^n.$ (Similarly for the other trig ratios.)
4. An equation is a statement that two algebraic expressions are equal. It may be true or false.
5. We can solve equations by trial and error, by using graphs, or by algebraic techniques.
6. To solve a trigonometric equation, we first isolate the trigonometric ratio on one side of the equation. We then use reference angles to find all the solutions between $0°$ and $360°.$
7. An equation that is true only for certain values of the variable, and false for others, is called a conditional equation. An equation that is true for all legitimate values of the variables is called an identity.
8. The expressions on either side of the equal sign in an identity are called equivalent expressions, because they have the same value for all values of the variable.
9. We often use identities to replace one form of an expression by a more useful form.
10. To check to whether an equation is an identity, we can compare graphs of $Y_1=$ (left side of the equation) and $Y_2=$ (right side of the equation). If the two graphs agree, the equation is an identity. If the two graphs are not the same, the equation is not an identity.
11. Pythagorean Identity. For any angle $\theta ,$${\cos }^2\theta +{\sin }^2\theta =1$Alternate forms:$$\begin{gathered} {\cos }^2\theta =1-{\sin }^2\theta \\ {\sin }^2\theta =1-{\cos }^2\theta \end{gathered}$$
12. Tangent Identity. For any angle not coterminal with $90°$ or $270°,$$\tan \theta ={\displaystyle \frac{\sin \theta }{\cos \theta }}$
13. To solve an equation involving more than one trig function, we use identities to rewrite the equation in terms of a single trig function.
14. To prove an identity, we write one side of the equation in equivalent forms until it is identical to the other side of the equation.