Stewart Precalc 6e Chapter 7 Test

Verify each identity. (a) \(\tan \theta \sin \theta + \cos \theta = \sec \theta\). (b) \(\tan x/(1 - \cos x) = \csc x (1 + \sec x)\). (c) \(\dfrac{2 \tan x}{1 + \tan^2 x} = \sin 2 x\).

Let \(x = 2 \sin \theta\), with \(-\dfrac{\pi}{2} < \theta < \dfrac{\pi}{2}\). Simplify the expression \(\dfrac{x}{\sqrt{4 - x^2}}\).

Find the exact value of each expression. (a) \(\sin 8^{\circ} \cos 22^{\circ} + \cos 8^{\circ} \sin 22^{\circ}\). (b) \(\sin 75^{\circ}\). (c) \(\sin\left(\dfrac{\pi}{12}\right)\).

For the angles \(\alpha\) and \(\beta\) shown in the figures, find \(\cos(\alpha + \beta)\).

(a) Write \(\sin 3 x \cos 5 x\) as a sum of trigonometric functions. (b) Write \(\sin 2 x - \sin 5 x\) as a product of trigonometric functions.

If \(\sin \theta = -\dfrac{4}{5}\) and \(\theta\) is in Quadrant III, find \(\tan\left(\dfrac{\theta}{2}\right)\).

Solve each trigonometric equation in \([0, 2 \pi)\), rounded to two decimal places. (a) \(3 \sin \theta - 1 = 0\). (b) \((2 \cos \theta - 1)(\sin \theta - 1) = 0\). (c) \(2 \cos^2 \theta + 5 \cos \theta + 2 = 0\). (d) \(\sin 2 \theta \cos \theta = 0\).

Find all solutions in \([0, 2 \pi)\), rounded to five decimal places: \(5 \cos 2 \theta = 2\).

Find the exact value of \(\cos(2 \arctan\left(\dfrac{9}{40}\right))\).

Rewrite the expression as an algebraic function of \(x\) and \(y\): \(\sin(\arccos x - \arctan y)\).

Find all solutions of the equation \(\cos(2 x) - \sin x = 0\) in the interval \([0, 2 \pi)\).