Chapter 6

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\csc x(\csc x-\sin x)+\frac{\sin x-\cos x}{\sin x}+\cot x=\csc ^{2} x$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$2 \sec ^{2} x+\tan ^{2} x=3$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$75^{\circ}$$

Write the trigonometric expression as an algebraic expression. $$\sin (\arcsin x+\arccos x)$$

Use a graphing utility to check your result graphically. $$\sin ^{4} x-\cos ^{4} x$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\cot x \tan x}{\cos x}=\sec x$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$4 \sin ^{2} x=2 \cos x+1$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$165^{\circ}$$

Write the trigonometric expression as an algebraic expression. $$\cos (\arccos x-\arcsin x)$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{1+\csc \theta}{\sec \theta}-\cot \theta=\cos \theta$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$\csc ^{2} x-3 \csc x=4$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$112^{\circ} 30^{\prime}$$

Write the trigonometric expression as an algebraic expression. $$\sin (\arctan 2 x-\arccos x)$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sec \theta \cot \theta=\csc \theta$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$\csc x+\cot x=1$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$67^{\circ} 30^{\prime}$$

Write the trigonometric expression as an algebraic expression. $$\cos (\arcsin x-\arctan 2 x)$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sin \theta \csc \theta-\sin ^{2} \theta=\cos ^{2} \theta$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$4 \sin x=\cos x-2$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{\pi}{8}$$

Find the value of the expression without using a calculator. $$\sin \left[\frac{\pi}{2}+\sin ^{-1}(-1)\right]$$

Perform the multiplication and use the fundamental identities to simplify. $$(\sin x+\cos x)^{2}$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}=2 \sec \theta$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$\frac{\cos x \cot x}{1-\sin x}=3$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{\pi}{12}$$

Find the value of the expression without using a calculator. $$\sin \left[\cos ^{-1}(-1)+\pi\right]$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\tan \theta}{1+\sec \theta}+\frac{1+\sec \theta}{\tan \theta}=2 \csc \theta$$

Use a graphing utility to approximate the solutions of the equation in the interval \([0,2 \pi)\) by collecting all terms on one side, graphing the new equation, and using the zero or root feature to approximate the \(x\) -intercepts of the graph. $$\frac{1+\sin x}{\cos ^{2} x}=2$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{7 \pi}{12}$$

Find the value of the expression without using a calculator. $$\sin \left(\sin ^{-1} 1+\cos ^{-1} 1\right)$$

Perform the multiplication and use the fundamental identities to simplify. $$(\csc x+1)(\csc x-1)$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\sin \beta}{1-\cos \beta}=\frac{1+\cos \beta}{\sin \beta}$$

(a) use a graphing utility to graph each function in the interval \([0,2 \pi),\) (b) write an equation whose solutions are the points of intersection of the graphs, and (c) use the intersect feature of the graphing utility to find the points of intersection (to four decimal places). $$y=\sin 2 x, \quad y=x^{2}-2 x$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{5 \pi}{8}$$

Find the value of the expression without using a calculator. $$\cos \left(\sin ^{-1} 1+\cos ^{-1} 0\right)$$

Perform the multiplication and use the fundamental identities to simplify. $$(5-5 \sin x)(5+5 \sin x)$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\cot \alpha}{\csc \alpha-1}=\frac{\csc \alpha+1}{\cot \alpha}$$

(a) use a graphing utility to graph each function in the interval \([0,2 \pi),\) (b) write an equation whose solutions are the points of intersection of the graphs, and (c) use the intersect feature of the graphing utility to find the points of intersection (to four decimal places). $$y=\cos x, \quad y=x+x^{2}$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\cos u=\frac{7}{25}, \quad 0

Use right triangles to evaluate the expression. $$\sin \left(\cos ^{-1} \frac{4}{5}+\sin ^{-1} \frac{3}{5}\right)$$

Perform the addition or subtraction and use the fundamental identities to simplify. $$\frac{1}{1+\cos x}+\frac{1}{1-\cos x}$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}}=\frac{1+\sin \alpha}{|\cos \alpha|}$$

(a) use a graphing utility to graph each function in the interval \([0,2 \pi),\) (b) write an equation whose solutions are the points of intersection of the graphs, and (c) use the intersect feature of the graphing utility to find the points of intersection (to four decimal places). $$y=\sin ^{2} x, \quad y=e^{x}-4 x$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\sin u=\frac{5}{13}, \quad \pi / 2

Use right triangles to evaluate the expression. $$\sin \left(\cos ^{-1} \frac{3}{5}-\sin ^{-1} \frac{5}{13}\right)$$

Perform the addition or subtraction and use the fundamental identities to simplify. $$\frac{1}{\sec x+1}-\frac{1}{\sec x-1}$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sqrt{\frac{1-\cos \beta}{1+\cos \beta}}=\frac{1-\cos \beta}{|\sin \beta|}$$

(a) use a graphing utility to graph each function in the interval \([0,2 \pi),\) (b) write an equation whose solutions are the points of intersection of the graphs, and (c) use the intersect feature of the graphing utility to find the points of intersection (to four decimal places). $$y=\cos ^{2} x, \quad y=e^{-x}+x-1$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\cot u=3, \quad \pi

Verify the identity. $$\sin \left(\frac{\pi}{2}+x\right)=\cos x$$