Chapter 6

Fill in the blank to complete the trigonometric formula. $$\sin (u-v)=$$ __________

Fill in the blank. The __________ solution of the equation \(2 \cos x-1=0\) is given by \(x=\frac{\pi}{3}+2 n \pi\) and \(x=\frac{5 \pi}{3}+2 n \pi,\) where \(n\) is an integer.

Fill in the blank to complete the trigonometric identity. $$\frac{1}{\tan u}=\text{_____}$$

Match each function with an equivalent expression. (a) \(\sin u\) (b) \(\cos u\) (c) \(\tan u\) (i) \(\frac{1}{\sec u}\) (ii) \(\frac{1}{\cot u}\) (iii) \(\frac{1}{\csc u}\)

Match each expression with an equivalent expression. (a) \(\sin ^{2} u\) (b) \(\sec ^{2} u\) (c) \(\csc ^{2} u\) (i) \(1+\cot ^{2} u\) (ii) \(1-\cos ^{2} u\) (iii) \(1+\tan ^{2} u\)

The equation \(\tan ^{2} x-5 \tan x+6=0\) is an equation of ___________ type.

Fill in the blank to complete the trigonometric identity. $$\frac{1}{\csc u}=\text{_____}$$

Fill in the blank to complete the trigonometric identity. $$\frac{\sin u}{\cos u}=\text{_____}$$

Is \(x=0\) a solution of the equation \(\cos x=0 ?\)

Fill in the blank to complete the trigonometric identity. $$\frac{1}{\sec u}=\text{_____}$$

To solve \(\sec x \sin ^{2} x=\sec x,\) do you divide each side by \(\sec x ?\)

Fill in the blank to complete the trigonometric formula. $$\cos (u-v)=$$ __________

Value is a solution of the equation. \(\tan x-\sqrt{3}=0\) (a) \(x=\frac{\pi}{3}\) (b) \(x=\frac{4 \pi}{3}\)

Fill in the blank to complete the trigonometric identity. $$\sin ^{2} u+ \text{_____} =1$$

Fill in the blank to complete the trigonometric formula. _______ \(=\frac{1-\cos 2 u}{1+\cos 2 u}\)

Value is a solution of the equation. \(\sec x-2=0\) (a) \(x=\frac{\pi}{3}\) (b) \(x=\frac{5 \pi}{3}\)

Fill in the blank to complete the trigonometric identity. $$\tan \left(\frac{\pi}{2}-u\right)=\text{_____}$$

Match each function with its double-angle formula. (a) \(\sin 2 u\) (b) \(\cos 2 u\) (c) \(\tan 2 u\) (i) \(1-2 \sin ^{2} u\) (ii) \(2 \sin u \cos u\) (iii) \(\frac{2 \tan u}{1-\tan ^{2} u}\)

Rewrite sin \(195^{\circ}\) so that you can use a sum formula.

Use the values to evaluate (if possible) all six trigonometric functions. $$\sin x=\frac{1}{2}, \quad \cos x=\frac{\sqrt{3}}{2}$$

Value is a solution of the equation. \(3 \tan ^{2} 2 x-1=0\) (a) \(x=\frac{\pi}{12}\) (b) \(x=\frac{5 \pi}{12}\)

Fill in the blank to complete the trigonometric identity. $$\cos (-u)=\text{_____}$$

Match each expression with its product-to-sum formula. (a) \(\sin u \cos v\) (b) \(\cos u \sin v\) (c) \(\cos u \cos v\) (i) \(\frac{1}{2}[\cos (u-v)+\cos (u+v)]\) (ii) \(\frac{1}{2}[\sin (u+v)+\sin (u-v)]\) (iii) \(\frac{1}{2}[\sin (u+v)-\sin (u-v)]\)

Rewrite \(\cos \frac{\pi}{12}\) so that you can use a difference formula.

Use the values to evaluate (if possible) all six trigonometric functions. $$\cos \theta=\frac{1}{2}, \quad \sin \theta=\frac{\sqrt{3}}{2}$$

Value is a solution of the equation. \(2 \cos ^{2} 4 x-1=0\) (a) \(x=\frac{\pi}{16}\) (b) \(x=\frac{3 \pi}{16}\)

Find the exact value of each expression. (a) \(\cos \left(240^{\circ}-0^{\circ}\right)\) (b) \(\cos 240^{\circ}-\cos 0^{\circ}\)

Use the values to evaluate (if possible) all six trigonometric functions. $$\cot \theta=-1, \quad \sin \theta=-\frac{\sqrt{2}}{2}$$

Value is a solution of the equation. \(2 \sin ^{2} x-\sin x-1=0\) (a) \(x=\frac{\pi}{2}\) (b) \(x=\frac{7 \pi}{6}\)

Is a graphical solution sufficient to verify a trigonometric identity?

Find the exact value of each expression. (a) \(\sin \left(405^{\circ}+120^{\circ}\right)\) (b) \(\sin 405^{\circ}+\sin 120^{\circ}\)

Value is a solution of the equation. \(\csc ^{4} x-4 \csc ^{2} x=0\) (a) \(x=\frac{\pi}{6}\) (b) \(x=\frac{5 \pi}{6}\)

Is a conditional equation true for all real values in its domain?

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\sin 2 x-\sin x=0$$

Find the exact value of each expression. (a) \(\sin \left(\frac{2 \pi}{3}+\frac{5 \pi}{6}\right)\) (b) \(\sin \frac{2 \pi}{3}+\sin \frac{5 \pi}{6}\)

Use the values to evaluate (if possible) all six trigonometric functions. $$\tan x=\frac{7}{24}, \quad \sec x=-\frac{25}{24}$$

Verify the identity. $$\sin t \csc t=1$$

Solving a Trigonometric Equation In Exercises \(11-16\) find all solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\sin x=0$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\sin 2 x+\cos x=0$$

Find the exact value of each expression. (a) \(\cos \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\) (b) \(\cos \frac{\pi}{4}+\cos \frac{\pi}{3}\)

Use the values to evaluate (if possible) all six trigonometric functions. $$\cot \phi=-5, \quad \sin \phi=\frac{\sqrt{26}}{26}$$

Verify the identity. $$\sec y \cos y=1$$

Solving a Trigonometric Equation In Exercises \(11-16\) fF\(\left[0^{\circ}, 360^{\circ}\right)\). $$\cos x=-1$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$4 \sin x \cos x=1$$

Find the exact values of the sine, cosine, and tangent of the angle. $$105^{\circ}=60^{\circ}+45^{\circ}$$

Use the values to evaluate (if possible) all six trigonometric functions. $$\sec \phi=-\frac{17}{15}, \quad \sin \phi=\frac{8}{17}$$

Verify the identity. $$\frac{\csc ^{2} x}{\cot x}=\csc x \sec x$$

Solving a Trigonometric Equation In Exercises \(11-16\) fF\(\left[0^{\circ}, 360^{\circ}\right)\). $$\cos x=-\frac{\sqrt{2}}{2}$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi) .\) If possible, find the exact solutions algebraically. $$\sin 2 x \sin x=\cos x$$

Find the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}=135^{\circ}+30^{\circ}$$