Chapter 5

Fill in the blank. \( \sin\left(u - v\right) \) =________

When solving a trigonometric equation, the preliminary goal is to ________ the trigonometric function involved in the equation.

Fill in the blank to complete the trigonometric identity. \( \dfrac{\sin u}{\cos u} \)= ________

The equation \( 2 \sin \theta + 1 = 0 \) has the solutions \( \theta = \frac{7\pi}{6} + 2n\pi \) and \( \theta = \frac{11\pi}{6} + 2n\pi \), which are called ________ solutions.

Fill in the blank to complete the trigonometric identity. \( \dfrac{1}{\csc u} \)= ________

Fill in the blank. \( \tan\left(u + v\right) \) =________

The equation \( 2 \tan^2 x - 3 \tan x + 1 = 0 \) is a trigonometric equation that is of ________ type.

In Exercises 3-8, fill in the blank to complete the trigonometric identity. \( \dfrac{1}{\cot u} \)= ________

Fill in the blank to complete the trigonometric identity. \( \dfrac{1}{\tan u} \)= ________

A solution of an equation that does not satisfy the original equation is called an ________ solution.

In Exercises 3-8, fill in the blank to complete the trigonometric identity. \( \dfrac{\cos u}{\sin u} \) = ________

Fill in the blank to complete the trigonometric identity. \( \dfrac{1}{\cos u} \)= ________

Fill in the blank. \( \cos\left(u - v\right) \) =________

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( 2 \cos x - 1 = 0 \) (a) \( x = \dfrac{\pi}{3} \) (b) \( x = \dfrac{5\pi}{3} \)

Fill in the blank. \( \tan \left(u - v\right) \) =________

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( \sec x - 2 = 0 \) (a) \( x = \dfrac{\pi}{3} \) (b) \( x = \dfrac{5\pi}{3} \)

In Exercises 3-8, fill in the blank to complete the trigonometric identity. \( \cos \left(\dfrac{\pi}{2} - u \right) \)= ________

In Exercises 7 - 12, find the exact value of each expression. (a) \( \cos\left(\dfrac{\pi}{4} + \dfrac{\pi}{3}\right) \) (b) \( \cos\dfrac{\pi}{4} + \cos \dfrac{\pi}{3} \)

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( 3 \tan^2 2x - 1 = 0 \) (a) \( x = \dfrac{\pi}{12} \) (b) \( x = \dfrac{5\pi}{12} \)

Fill in the blank to complete the trigonometric identity. \( \sin \left(\dfrac{\pi}{2} - u\right) \)= ________

In Exercises 7 - 12, find the exact value of each expression. (a) \( \sin\left(\dfrac{3\pi}{4} + \dfrac{5\pi}{6}\right) \) (b) \( \sin \dfrac{3\pi}{4} + \sin \dfrac{5\pi}{6} \)

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( 2 \cos^2 4x - 1 = 0 \) (a) \( x = \dfrac{\pi}{16} \) (b) \( x = \dfrac{3\pi}{16} \)

In Exercises 3-8, fill in the blank to complete the trigonometric identity. \( \sec (-u) \)= ________

Fill in the blank to complete the trigonometric identity. \( \sec\left(\dfrac{\pi}{2} - u \right) \)= ________

In Exercises 7 - 12, find the exact value of each expression. (a) \( \sin\left(\dfrac{7\pi}{6} + \dfrac{\pi}{3}\right) \) (b) \( \sin\dfrac{7\pi}{6} - \cos \dfrac{\pi}{3} \)

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( 2 \sin^2 x - \sin x - 1 = 0 \) (a) \( x = \dfrac{\pi}{2} \) (b) \( x = \dfrac{7\pi}{6} \)

In Exercises 9-50, verify the identity \( \tan t \cot t = 1 \)

Fill in the blank to complete the trigonometric identity. \( \cos(-u) \)= ________

In Exercises 7 - 12, find the exact value of each expression. (a) \( \cos\left(120^\circ + 45^\circ\right) \) (b) \( \cos 120^\circ + 45^\circ \)

In Exercises 5-10, verify that the \( x \)-values are solutions of the equation. \( \csc^4 x - 4 \csc^2 x = 0 \) (a) \( x = \dfrac{\pi}{6} \) (b) \( x = \dfrac{5\pi}{6} \)

In Exercises 9-50, verify the identity \( \sec y \cos y = 1 \)

In Exercises 7 - 12, find the exact value of each expression. (a) \( \sin\left(135^\circ - 30^\circ\right) \) (b) \( \sin 135^\circ - 30^\circ \)

In Exercises 11-24, solve the equation. \( 2 \cos x + 1 = 0 \)

In Exercises 9-50, verify the identity \( \cot^2 y (\sec^2 y - 1) = 1 \)

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \sin x = \dfrac{1}{2} \), \( \cos x = \dfrac{\sqrt{3}}{2} \)

In Exercises 7 - 12, find the exact value of each expression. (a) \( \sin\left(315^\circ - 60^\circ\right) \) (b) \( \sin 315^\circ - 60^\circ \)

In Exercises 11-24, solve the equation. \( 2 \sin x + 1 = 0 \)

In Exercises 9-50, verify the identity \( \cos x + \sin x \tan x = \sec x \)

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \tan x = \dfrac{\sqrt{3}}{3} \), \( \cos x = - \dfrac{\sqrt{3}}{2} \)

In Exercises 13 - 28, find the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{11\pi}{12} = \dfrac{3\pi}{4} + \dfrac{\pi}{6} \)

In Exercises 11-24, solve the equation. \( \sqrt{3} \csc x - 2 = 0 \)

In Exercises 9-50, verify the identity \( (1 + \sin \alpha) (1 - \sin \alpha) = \cos^2 \alpha \)

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \sec \theta = \sqrt{2} \), \( \sin \theta = - \dfrac{\sqrt{3}}{2} \)

In Exercises 13 - 28, find the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{7\pi}{12} = \dfrac{\pi}{3} + \dfrac{\pi}{4} \)

In Exercises 11-24, solve the equation. \( \tan x + \sqrt{3} = 0 \)

In Exercises 9-50, verify the identity \( \cos^2 \beta - \sin^2 \beta = 2 \cos^2 \beta - 1 \)

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \csc \theta = \dfrac{25}{7} \), \( \tan \theta = \dfrac{7}{24} \)

In Exercises 13 - 28, find the exact values of the sine, cosine, and tangent of the angle. \( \dfrac{17\pi}{12} = \dfrac{9\pi}{4} - \dfrac{5\pi}{6} \)

In Exercises 11-24, solve the equation. \( 3 \sec^2 x - 4 = 0 \)

In Exercises 9-50, verify the identity \( \cos^2 \beta - \sin^2 \beta = 1 - 2 \sin^2 \beta - 1 \)