1.
[latex]\sin^2 x \cos y[/latex]
Chapter 5: Equations and Identities
For Problems 1–4, evaluate the expressions for [latex]x = 120°,~ y = 225°,[/latex] and [latex]z = 90°{.}[/latex] Give exact values for your answers.
[latex]\sin^2 x \cos y[/latex]
[latex]\sin z - \dfrac{1}{2} \sin y[/latex]
[latex]\tan (z - x) \cos (y - z)[/latex]
[latex]\dfrac{\tan^2 x}{2 \cos y}[/latex]
For Problems 5–8, evaluate the expressions using a calculator. Are they equal?
For Problems 9–12, simplify the expression.
[latex]3\sin x - 2\sin x \cos y + 2\sin x - \cos y[/latex]
[latex]\cos t + 3\cos 3t - 3\cos t - 2\cos 3t[/latex]
[latex]6 \tan^2 \theta + 2\tan \theta - (4\tan \theta )^2[/latex]
[latex]\sin \theta (2\cos \theta - 2) + \sin \theta (1 - \sin \theta)[/latex]
For Problems 13–16, decide whether or not the expressions are equivalent. Explain.
[latex]\cos \theta + \cos 2\theta;~~\cos 3\theta[/latex]
[latex]1 + \sin^2 x;~~(1 + \sin x)^2[/latex]
[latex]3\tan^2 t - \tan^2 t;~~2\tan^2 t[/latex]
[latex]\cos 4\theta;~~2\cos 2\theta[/latex]
For Problems 17–20, multiply or expand.
[latex](\cos \alpha + 2)(2\cos \alpha - 3)[/latex]
[latex](1 - 3\tan \beta)^2[/latex]
[latex](\tan \phi - \cos \phi)^2 = 0[/latex]
[latex](\sin \rho - 2\cos \rho)(\sin \rho + \cos \rho)[/latex]
For Problems 21–24, factor the expression.
[latex]12\sin 3x - 6\sin 2x[/latex]
[latex]2\cos^2 \beta + \cos \beta[/latex]
[latex]1 - 9\tan^2 \theta[/latex]
[latex]\sin^2 \phi - \sin \phi \tan \phi - 2\tan^2 \phi[/latex]
For Problems 25–30, reduce the fraction.
[latex]\dfrac{\cos^2 \alpha - \sin^2 \alpha}{\cos \alpha - \sin \alpha}[/latex]
[latex]\dfrac{1 - \tan^2 \theta}{1 - \tan \theta}[/latex]
[latex]\dfrac{3\cos x + 9}{2\cos x + 6}[/latex]
[latex]\dfrac{5\sin \theta - 10}{\sin^2 \theta - 4}[/latex]
[latex]\dfrac{3\tan^2 C - 12}{\tan^2 C - 4\tan C + 4}[/latex]
[latex]\dfrac{\tan^2 \beta - \tan \beta - 6}{\tan \beta - 3}[/latex]
For Problems 31–32, use a graph to solve the equation for [latex]0° \le x \lt 360°{.}[/latex] Check your solutions by substitution.
[latex]8\cos x - 3 = 2[/latex]
[latex]6\tan x - 2 = 8[/latex]
For Problems 33–40, find all solutions between [latex]0°[/latex] and [latex]360°{.}[/latex] Give exact answers.
[latex]2\cos^2 \theta + \cos \theta = 0[/latex]
[latex]\sin^2 \alpha - \sin \alpha = 0[/latex]
[latex]2\sin^2 x - \sin x - 1 = 0[/latex]
[latex]\cos^2 B + 2\cos B + 1 = 0[/latex]
[latex]\tan^2 x = \dfrac{1}{3}[/latex]
[latex]\tan^2 t - \tan t = 0[/latex]
[latex]6\cos^2 \alpha - 3\cos \alpha - 3 = 0[/latex]
[latex]2\sin^2 \theta + 4\sin \theta + 2 = 0[/latex]
For Problems 41–44, solve the equation for [latex]0° \le x \lt 360°{.}[/latex] Round your answers to two decimal places.
[latex]2 - 5\tan \theta = -6[/latex]
[latex]3 + 5\cos \theta = 4[/latex]
[latex]3\cos^2 x + 7\cos x = 0[/latex]
[latex]8 - 9\sin^2 x = 0[/latex]
A light ray passes from glass to water, with a [latex]37°[/latex] 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 [latex]76°[/latex] 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.
[latex]\cos x\tan x = \sin x[/latex]
[latex]\sin \theta = 1 - \cos \theta[/latex]
[latex]\tan \phi + \tan \phi = \tan 2\phi[/latex]
[latex]\tan^2 x = \dfrac{\sin^2 x}{1 - \sin^2 x}[/latex]
For Problems 51–54, use graphs to decide which of the following equations are identities.
[latex]\cos 2\theta = 2 \cos \theta[/latex]
[latex]\cos (x - 90°) = \sin x[/latex]
[latex]\sin 2x = 2\sin x \cos x[/latex]
[latex]\cos (\theta + 90°) = \cos \theta - 1[/latex]
For Problems 55–58, show that the equation is an identity by transforming the left side into the right side.
[latex]\dfrac{1 - \cos^2 \alpha}{\tan \alpha} = \sin \alpha \cos \alpha[/latex]
[latex]\cos^2 \beta \tan^2 \beta + \cos^2 \beta = 1[/latex]
[latex]\dfrac{\tan \theta - \sin \theta \cos \theta}{\sin \theta \cos \theta} = \sin \theta[/latex]
[latex]\tan \phi - \dfrac{\sin^2 \phi}{\tan \phi} = \tan \phi \sin^2 \phi[/latex]
For Problems 59–62, simplify, using identities as necessary.
[latex]\tan \theta + \dfrac{\cos \theta}{\sin \theta}[/latex]
[latex]\dfrac{1 - 2\cos^2 \beta}{\sin \beta \cos \beta} + \dfrac{\cos \beta}{\sin \beta}[/latex]
[latex]\dfrac{1}{1 - \sin^2 v} - \tan^2 v[/latex]
[latex]\cos u + (\sin u)(\tan u)[/latex]
For Problems 63–66, evaluate the expressions without using a calculator.
[latex]\sin 137° - \tan 137° \cdot \cos 137°[/latex]
[latex]\cos^2 8° + \cos 8° \cdot \tan 8° \cdot \sin 8°[/latex]
[latex]\dfrac{1}{\cos^2 54°} - \tan^2 54°[/latex]
[latex]\dfrac{2}{\cos^2 7°} - 2\tan^2 7°[/latex]
For Problems 67–70, use identities to rewrite each expression.
Write [latex]\tan^2 \beta + 1[/latex] in terms of [latex]\cos^2 \beta{.}[/latex]
Write [latex]2\sin^2 t + \cos t[/latex] in terms of [latex]\cos t{.}[/latex]
Write [latex]\dfrac{\cos x}{\tan x}[/latex] in terms of [latex]\sin x{.}[/latex]
Write [latex]\tan^2 \beta + 1[/latex] in terms of [latex]\cos^2 \beta{.}[/latex]
For Problems 71–74, find the values of the three trigonometric functions.
[latex]7\tan \beta - 4 = 2, ~~ 180° \lt \beta \lt 270°[/latex]
[latex]3\tan C + 5 = 3, ~~-90° \lt C \lt 0°[/latex]
[latex]5\cos \alpha + 3 = 1, ~~ 90° \lt \alpha \lt 180°[/latex]
[latex]3\sin \theta + 2 = 4, ~~ 90° \lt \beta \lt 180°[/latex]
For Problems 75–82, solve the equation for [latex]0° \le x \lt 360°{.}[/latex] Round angles to three decimal places if necessary.
[latex]\sin w + 1 = \cos^2 w[/latex]
[latex]\cos^2 \phi - \cos \phi - \sin^2 \phi = 0[/latex]
[latex]\cos x + \sin x = 0[/latex]
[latex]3\sin \theta = \sqrt{3} \cos \theta[/latex]
[latex]2\sin \beta - \tan \beta = 0[/latex]
[latex]6\tan \theta \cos \theta + 6 = 0[/latex]
[latex]\cos^2 t - \sin^2 t = 1[/latex]
[latex]5\cos^2 \beta - 5\sin^2 \beta = -5[/latex]