###### Pythagorean Identity

For any angle \(\theta\text{,}\)

Alternate forms:

- 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 \(\sin (X + Y)\) indicate function notation, not multiplication.
- We write \(\cos^2 \theta\) to denote \((\cos \theta)^2\text{,}\) and \(\cos^n \theta\) to denote \((\cos \theta)^n\text{.}\) (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 \(0\degree\) and \(360\degree\text{.}\)
- 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 \(Y_1 = \) (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.
###### Pythagorean Identity

For any angle \(\theta\text{,}\)

\begin{equation*} \cos^2 \theta + \sin^2 \theta = 1 \end{equation*}Alternate forms:

\begin{equation*} \begin{aligned}[t] \cos^2 \theta \amp = 1 - \sin^2 \theta\\ \sin^2 \theta \amp = 1 - \cos^2 \theta \\ \end{aligned} \end{equation*}###### Tangent Identity

For any angle not coterminal with \(90\degree\) or \(270\degree\text{,}\)

\begin{equation*} \tan \theta = \dfrac{\sin \theta}{\cos \theta} \end{equation*}- 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.

For Problems 1–4, evaluate the expressions for \(x = 120\degree,~ y = 225\degree,~ \text{and}~ z = 90\degree\text{.}\) Give exact values for your answers.

\(\sin^2 x \cos y\)

\(\sin z - \dfrac{1}{2} \sin y\)

\(\tan (z - x) \cos (y - z)\)

\(\dfrac{\tan^2 x}{2 \cos y}\)

For Problems 5–8, evaluate the expressions using a calculator. Are they equal?

- \(\sin (20\degree + 40\degree)\)
- \(\sin 20\degree + \sin 40\degree\)

- \(\cos^2 70\degree - \sin^2 70\degree\)
- \(\cos (2\cdot 70\degree)\)

- \(\dfrac{\sin 55\degree}{\cos 55\degree}\)
- \(\tan 55\degree\)

- \(\tan 80\degree - \tan 10\degree\)
- \(\tan (80\degree - 10\degree)\)

For Problems 9–12, simplify the expression.

\(3\sin x - 2\sin x \cos y + 2\sin x - \cos y\)

\(\cos t + 3\cos 3t - 3\cos t - 2\cos 3t\)

\(6 \tan^2 \theta + 2\tan \theta - (4\tan \theta )^2\)

\(\sin \theta (2\cos \theta - 2) + \sin \theta (1 - \sin \theta)\)

For Problems 13–16, decide whether or not the expressions are equivalent. Explain.

\(\cos \theta + \cos 2\theta;~~\cos 3\theta\)

\(1 + \sin^2 x;~~(1 + \sin x)^2\)

\(3\tan^2 t - \tan^2 t;~~2\tan^2 t\)

\(\cos 4\theta;~~2\cos 2\theta\)

For Problems 17–20, multiply or expand.

\((\cos \alpha + 2)(2\cos \alpha - 3)\)

\((1 - 3\tan \beta)^2\)

\((\tan \phi - \cos \phi)^2 = 0\)

\((\sin \rho - 2\cos \rho)(\sin \rho + \cos \rho)\)

For Problems 21–24, factor the expression.

\(12\sin 3x - 6\sin 2x\)

\(2\cos^2 \beta + \cos \beta\)

\(1 - 9\tan^2 \theta\)

\(\sin^2 \phi - \sin \phi \tan \phi - 2\tan^2 \phi\)

For Problems 25–30, reduce the fraction.

\(\dfrac{\cos^2 \alpha - \sin^2 \alpha}{\cos \alpha - \sin \alpha}\)

\(\dfrac{1 - \tan^2 \theta}{1 - \tan \theta}\)

\(\dfrac{3\cos x + 9}{2\cos x + 6}\)

\(\dfrac{5\sin \theta - 10}{\sin^2 \theta - 4}\)

\(\dfrac{3\tan^2 C - 12}{\tan^2 C - 4\tan C + 4}\)

\(\dfrac{\tan^2 \beta - \tan \beta - 6}{\tan \beta - 3}\)

For Problems 31–32, use a graph to solve the equation for \(0\degree \le x \lt 360\degree\text{.}\) Check your solutions by substitution.

\(8\cos x - 3 = 2\)

\(6\tan x - 2 = 8\)

For Problems 33–40, find all solutions between \(0\degree\) and \(360\degree\text{.}\) Give exact answers.

\(2\cos^2 \theta + \cos \theta = 0\)

\(\sin^2 \alpha - \sin \alpha = 0\)

\(2\sin^2 x - \sin x - 1 = 0\)

\(\cos^2 B + 2\cos B + 1 = 0\)

\(\tan^2 x = \dfrac{1}{3}\)

\(\tan^2 t - \tan t = 0\)

\(6\cos^2 \alpha - 3\cos \alpha - 3 = 0\)

\(2\sin^2 \theta + 4\sin \theta + 2 = 0\)

For Problems 41–44, solve the equation for \(0\degree \le x \lt 360\degree\text{.}\) Round your answers to two decimal places.

\(2 - 5\tan \theta = -6\)

\(3 + 5\cos \theta = 4\)

\(3\cos^2 x + 7\cos x = 0\)

\(8 - 9\sin^2 x = 0\)

A light ray passes from glass to water, with a \(37\degree\) angle of incidence. What is the angle of refraction? The index of refraction from water to glass is 1.1.

A light ray passes from glass to water, with a \(76\degree\) angle of incidence. What is the angle of refraction? The index of refraction from water to glass is 1.1.

For Problems 47–50, decide which of the following equations are identities. Explain your reasoning.

\(\cos x\tan x = \sin x\)

\(\sin \theta = 1 - \cos \theta\)

\(\tan \phi + \tan \phi = \tan 2\phi\)

\(\tan^2 x = \dfrac{\sin^2 x}{1 - \sin^2 x}\)

For Problems 51–54, use graphs to decide which of the following equations are identities.

\(\cos 2\theta = 2 \cos \theta\)

\(\cos (x - 90\degree) = \sin x\)

\(\sin 2x = 2\sin x \cos x\)

\(\cos (\theta + 90\degree) = \cos \theta - 1\)

For Problems 55–58, show that the equation is an identity by transforming the left side into the right side.

\(\dfrac{1 - \cos^2 \alpha}{\tan \alpha} = \sin \alpha \cos \alpha\)

\(\cos^2 \beta \tan^2 \beta + \cos^2 \beta = 1\)

\(\dfrac{\tan \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta} = \sin \theta\)

\(\tan \phi - \dfrac{\sin^2 \phi}{\tan \phi} = \tan \phi \sin^2 \phi\)

For Problems 59–62, simplify, using identities as necessary.

\(\tan \theta + \dfrac{\cos \theta}{\sin \theta}\)

\(\dfrac{1 - 2\cos^2 \beta}{\sin \beta \cos \beta} + \dfrac{\cos \beta}{\sin \beta}\)

\(\dfrac{1}{1 - \sin^2 v} - \tan^2 v\)

\(\cos u + (\sin u)(\tan u)\)

For Problems 63–66, evaluate the expressions without using a calculator.

\(\sin 137\degree - \tan 137\degree \cdot \cos 137\degree\)

\(\cos^2 8\degree + \cos 8\degree \cdot \tan 8\degree \cdot \sin 8\degree\)

\(\dfrac{1}{\cos^2 54\degree} - \tan^2 54\degree\)

\(\dfrac{2}{\cos^2 7\degree} - 2\tan^2 7\degree\)

For Problems 67–70,use identities to rewrite each expression.

Write \(\tan^2 \beta + 1\) in terms of \(\cos^2 \beta\text{.}\)

Write \(2\sin^2 t + \cos t\) in terms of \(\cos t\text{.}\)

Write \(\dfrac{\cos x}{\tan x}\) in terms of \(\sin x\text{.}\)

Write \(\tan^2 \beta + 1\) in terms of \(\cos^2 \beta\text{.}\)

For Problems 71–74, find the values of the three trigonometric functions.

\(7\tan \beta - 4 = 2, ~~ 180\degree \lt \beta \lt 270\degree\)

\(3\tan C + 5 = 3, ~~-90\degree \lt C \lt 0\degree\)

\(5\cos \alpha + 3 = 1, ~~ 90\degree \lt \alpha \lt 180\degree\)

\(3\sin \theta + 2 = 4, ~~ 90\degree \lt \beta \lt 180\degree\)

For Problems 75–82, solve the equation for \(0\degree \le x \lt 360\degree\text{.}\) Round angles to three decimal places if necessary.

\(\sin w + 1 = \cos^2 w\)

\(\cos^2 \phi - \cos \phi - \sin^2 \phi = 0\)

\(\cos x + \sin x = 0\)

\(3\sin \theta = \sqrt{3} \cos \theta\)

\(2\sin \beta - \tan \beta = 0\)

\(6\tan \theta \cos \theta + 6 = 0\)

\(\cos^2 t - \sin^2 t = 1\)

\(5\cos^2 \beta - 5\sin^2 \beta = -5\)

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