Chapter 5: Equations and Identities
Chapter 5 Summary and Review
Key Concepts
- 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.
- The parentheses in an expression such as [latex]\sin (X + Y)[/latex] indicate function notation, not multiplication.
- We write [latex]\cos^2 \theta[/latex] to denote [latex](\cos \theta)^2{,}[/latex] and [latex]\cos^n \theta[/latex] to denote [latex](\cos \theta)^n{.}[/latex] (Similarly for the other trig ratios.)
- An equation is a statement that two algebraic expressions are equal. It may be true or false.
- We can solve equations by trial and error, by using graphs, or by algebraic techniques.
- 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 [latex]0°[/latex] and [latex]360°{.}[/latex]
- 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.
- 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.
- We often use identities to replace one form of an expression by a more useful form.
- To check to whether an equation is an identity, we can compare graphs of [latex]Y_1 =[/latex] (left side of the equation) and [latex]Y_2 =[/latex] (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.
- Pythagorean Identity. For any angle [latex]\theta{,}[/latex][latex]\cos^2 \theta + \sin^2 \theta = 1[/latex]Alternate forms:[latex]\cos^2 \theta = 1 - \sin^2 \theta\\ \sin^2 \theta = 1 - \cos^2 \theta[/latex]
- Tangent Identity. For any angle not coterminal with [latex]90°[/latex] or [latex]270°{,}[/latex][latex]\tan \theta = \dfrac{\sin \theta}{\cos \theta}[/latex]
- To solve an equation involving more than one trig function, we use identities to rewrite the equation in terms of a single trig function.
- 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.