Chapter 10

Suppose you are solving a trigonometric equation for solutions in \([0,2 \pi)\) and your work leads to $$ 2 x=\frac{2 \pi}{3}, 2 \pi, \frac{8 \pi}{3} $$ What are the corresponding values of \(x ?\)

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \cos x+1=0$$

Complete each statement, or answer the question. For a function to have an inverse, it must be ____.

Fill in the blank(s) to complete each fundamental identity. \(\sin ^{2} x+\cos ^{2} x=\) ________

Match each expression with the correct expression to form an identity. $$\cos (x+y)= \text {_____} $$ A. \(\cos x \cos y+\sin x \sin y\) B. \(\sin x \sin y-\cos x \cos y\) C. \(\sin x \cos y+\cos x \sin y\) D. \(\sin x \cos y-\cos x \sin y\) E. \(\cos x \sin y-\sin x \cos y\) F. \(\cos x \cos y-\sin x \sin y\)

Suppose you are solving a trigonometric equation to find solutions in \(\left[0^{\circ}, 360^{\circ}\right)\) and your work leads to $$ \frac{1}{3} \theta=45^{\circ}, 60^{\circ}, 75^{\circ}, 90^{\circ} $$ What are the corresponding values of \(\theta ?\)

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \sin x+1=0$$

Complete each statement, or answer the question. The domain of \(y=\arcsin x\) equals the ____ of \(y=\sin x\).

Fill in the blank(s) to complete each fundamental identity. \(\tan ^{2} x+1=\) ________

Match each expression with the correct expression to form an identity. $$\cos (x-y)= \text {_____} $$ A. \(\cos x \cos y+\sin x \sin y\) B. \(\sin x \sin y-\cos x \cos y\) C. \(\sin x \cos y+\cos x \sin y\) D. \(\sin x \cos y-\cos x \sin y\) E. \(\cos x \sin y-\sin x \cos y\) F. \(\cos x \cos y-\sin x \sin y\)

Solve each equation over the interval \([0,2 \pi)\) $$\sin \frac{x}{2}=\cos \frac{x}{2}$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$5 \sin x-6=0$$

Match each expression in Column I with its value in Column \(I I\). A. \(\frac{1}{2}\) B. \(\frac{\sqrt{2}}{2}\) c. \(\frac{\sqrt{3}}{2}\) D. \(-\sqrt{3}\) E. \(\frac{\sqrt{3}}{3}\) F. \(\sqrt{3}\) $$2 \sin 22.5^{\circ} \cos 22.5^{\circ}$$

Complete each statement, or answer the question. \(y=\cos ^{-1} x\) means that \(x=\) _____, for \(0 \leq y \leq \pi\).

Fill in the blank(s) to complete each fundamental identity. \(1+\cot ^{2} x=\) ________

Match each expression with the correct expression to form an identity. $$\sin (x+y)= \text {_____} $$ A. \(\cos x \cos y+\sin x \sin y\) B. \(\sin x \sin y-\cos x \cos y\) C. \(\sin x \cos y+\cos x \sin y\) D. \(\sin x \cos y-\cos x \sin y\) E. \(\cos x \sin y-\sin x \cos y\) F. \(\cos x \cos y-\sin x \sin y\)

Solve each equation over the interval \([0,2 \pi)\) $$\sec \frac{x}{2}=\cos \frac{x}{2}$$

Match each expression in Column I with its value in Column \(I I\). A. \(\frac{1}{2}\) B. \(\frac{\sqrt{2}}{2}\) c. \(\frac{\sqrt{3}}{2}\) D. \(-\sqrt{3}\) E. \(\frac{\sqrt{3}}{3}\) F. \(\sqrt{3}\) $$\cos ^{2} \frac{\pi}{6}-\sin ^{2} \frac{\pi}{6}$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$3 \cos x+5=0$$

Complete each statement, or answer the question. The point \(\left(\frac{\pi}{4}, 1\right)\) lies on the graph of \(y=\tan x .\) Thus, the point _____ lies on the graph of ____.

Fill in the blank(s) to complete each fundamental identity. \(\csc x=\frac{1}{\underline{\phantom{xx}}}\)

Match each expression with the correct expression to form an identity. $$\sin (x-y)= \text {_____} $$ A. \(\cos x \cos y+\sin x \sin y\) B. \(\sin x \sin y-\cos x \cos y\) C. \(\sin x \cos y+\cos x \sin y\) D. \(\sin x \cos y-\cos x \sin y\) E. \(\cos x \sin y-\sin x \cos y\) F. \(\cos x \cos y-\sin x \sin y\)

Solve each equation over the interval \([0,2 \pi)\) $$\sin ^{2} \frac{x}{2}-1=0$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \tan x+1=-1$$

Complete each statement, or answer the question. If a function \(f\) has an inverse and \(f(\pi)=-1,\) then \(f^{-1}(-1)=\) _____.

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin \frac{\pi}{12}$$

Solve each equation over the interval \([0,2 \pi)\) $$\sin x \cos x=\frac{1}{4}$$

Match each expression in Column I with its value in Column \(I I\). A. \(\frac{1}{2}\) B. \(\frac{\sqrt{2}}{2}\) c. \(\frac{\sqrt{3}}{2}\) D. \(-\sqrt{3}\) E. \(\frac{\sqrt{3}}{3}\) F. \(\sqrt{3}\) $$\frac{2 \tan \frac{\pi}{3}}{1-\tan ^{2} \frac{\pi}{3}}$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \cot x+1=-1$$

Complete each statement, or answer the question. How can the graph of \(f^{-1}\) be sketched if the graph of \(f\) is known?

Fill in the blank(s) to complete each fundamental identity. \(\frac{\sin x}{\cos x}=\) _______

Use identities to find the exact value of each expression. Do not use a calculator. $$\tan \frac{\pi}{12}$$

Solve each equation over the interval \([0,2 \pi)\) $$\sin 2 x=2 \cos ^{2} x$$

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \cos x+5=6$$

Write short answers and fill in the blanks. Consider the inverse sine function \(y=\sin ^{-1} x,\) or \(y=\arcsin x\) (a) What is its domain? (b) What is its range? (c) For this function, as \(x\) increases, \(y\) increases. Therefore, it is a(n) _____ function. (d) Why is arcsin (-2) not defined?

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\cos (-4.38)$$

Use identities to find the exact value of each expression. Do not use a calculator. $$\tan \left(-\frac{5 \pi}{12}\right)$$

Solve each equation over the interval \([0,2 \pi)\) $$\csc ^{2} \frac{x}{2}=2 \sec x$$

Determine whether the positive or negative square root should be chosen in each application of a half-angle identity. $$\cos 58^{\circ}=\pm \sqrt{\frac{1+\cos 116^{\circ}}{2}}$$

Write short answers and fill in the blanks. Consider the inverse cosine function \(y=\cos ^{-1} x,\) or \(y=\arccos x\) (a) What is its domain? (b) What is its range? (c) For this function, as \(x\) increases, \(y\) decreases. Therefore, it is a(n) _____ function. (d) The equation \(\cos \left(-\frac{4 \pi}{3}\right)=-\frac{1}{2}\) is a true statement. Why is arccos \(\left(-\frac{1}{2}\right)\) not equal to \(-\frac{4 \pi}{3} ?\)

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \sin x+3=4$$

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\cos (-5.46)$$

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin \left(-\frac{5 \pi}{12}\right)$$

Solve each equation over the interval \([0,2 \pi)\) $$\cos x-1=\cos 2 x$$

Write short answers and fill in the blanks. Consider the inverse tangent function, \(y=\tan ^{-1} x\) or \(y=\arctan x\) (a) What is its domain? (b) What is its range? (c) For this function, as \(x\) increases, \(y\) increases. Therefore, it is a(n) _____ function. (d) Is there any real number \(x\) for which arctan \(x\) is not defined? If so, what is it (or what are they)?

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \csc x+4=\csc x+6$$

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\sin (-0.5)$$

Use identities to find the exact value of each expression. Do not use a calculator. $$\sin \left(\frac{13 \pi}{12}\right)$$

Solve each equation over the interval \([0,2 \pi)\) $$1-\sin x=\cos 2 x$$

Determine whether the positive or negative square root should be chosen in each application of a half-angle identity. $$\sin \left(-10^{\circ}\right)=\pm \sqrt{\frac{1-\cos \left(-20^{\circ}\right)}{2}}$$