Chapter 6

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. Use a graphing utility to verify your results. $$y=|2 x-9|-5$$

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln (1+\sin x)-\ln |\sec x|$$

The monthly unit sales \(U\) (in thousands) of lawn mowers are approximated by $$U=74.50-43.75 \cos \frac{\pi t}{6}$$where \(t\) is the time (in months), with \(t=1\) corresponding to January. Determine the months in which unit sales exceed 100,000

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. Use a graphing utility to verify your results. $$y=2 x \sqrt{x+7}$$

Rewrite the expression as a single logarithm and simplify the result. (Hint: Begin by using the properties of logarithms.) $$\ln |\cot t|+\ln \left(1+\tan ^{2} t\right)$$

The monthly unit sales \(U\) (in hundreds) of skis for a chain of sports stores are approximated by \(U=58.3+32.5 \cos \frac{\pi t}{6}\) where \(t\) is the time (in months), with \(t=1\) corresponding to January. Determine the months in which unit sales exceed 7500

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

Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\csc ^{2} \theta-\cot ^{2} \theta=1$$ (a) \(\theta=132^{\circ}\) (b) \(\theta=\frac{2 \pi}{7}\)

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\tan \frac{u}{2}=\csc u-\cot u$$

Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\tan ^{2} \theta+1=\sec ^{2} \theta$$ (a) \(\theta=346^{\circ}\) (b) \(\theta=3.1\)

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha=\cos 4 \alpha$$

Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\cos \left(\frac{\pi}{2}-\theta\right)=\sin \theta$$ (a) \(\theta=80^{\circ}\) (b) \(\theta=0.8\)

Harmonic Motion A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by \(y=\frac{1}{4}(\cos 8 t-3 \sin 8 t)\) where \(y\) is the displacement (in meters) and \(t\) is the time (in seconds). Find the times at which the weight is at the point of equilibrium \((y=0)\) for \(0 \leq t \leq 1\)

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$

Use the table feature of a graphing utility to demonstrate the identity for each value of \(\theta\) $$\sin (-\theta)=-\sin \theta$$ (a) \(\theta=250^{\circ}\) (b) \(\theta=\frac{\pi}{4}\)

Damped Harmonic Motion The displacement from equilibrium of a weight oscillating on the end of a spring is given by \(y=1.56 e^{-0.22 t} \cos 4.9 t,\) where \(y\) is the displacement (in feet) and \(t\) is the time (in seconds). Use a graphing utility to graph the displacement function for \(0 \leq t \leq 10 .\) Find the time beyond which the displacement does not exceed 1 foot from equilibrium.

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$(\sin x+\cos x)^{2}=1+\sin 2 x$$

The forces acting on an object weighing \(W\) units on an inclined plane positioned at an angle of \(\theta\) with the horizontal are modeled by $$\mu W \cos \theta=W \sin \theta$$ where \(\mu\) is the coefficient of friction (see figure). Solve the equation for \(\mu\) and simplify the result.

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sin \frac{\alpha}{3} \cos \frac{\alpha}{3}=\frac{1}{2} \sin \frac{2 \alpha}{3}$$

Consider the function \(f(x)=3 \sin (0.6 x-2)\) (a) Find the zero of \(f\) in the interval [0,6] (b) A quadratic approximation of \(f\) near \(x=4\) is \(g(x)=-0.45 x^{2}+5.52 x-13.70\) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Describe the result. (c) Use the Quadratic Formula to approximate the zeros of \(g .\) Compare the zero in the interval [0,6] with the result of part (a).

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\sin 3 y}{\sin y}=1-2 \sin ^{2} y+2 \cos ^{2} y$$

The area of a rectangle inscribed in one arc of the graph of \(y=\cos x\) (see figure) is given by \(A=2 x \cos x, \quad 0 \leq x \leq \frac{\pi}{2}\). (a) Use a graphing utility to graph the area function and approximate the area of the largest inscribed rectangle. (b) Determine the values of \(x\) for which \(A \geq 1\)

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\cos 3 \beta}{\cos \beta}=1-4 \sin ^{2} \beta$$

Determine whether the statement is true or false. Justify your answer. $$\sin \theta \csc \theta=1$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\csc \frac{u}{2}=\pm \sqrt{\frac{2 \csc u}{\csc u-\cot u}}$$

Determine whether the statement is true or false. Justify your answer. $$\cos \theta \sec \phi=1$$

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sec \frac{u}{2}=\pm \sqrt{\frac{2 \tan u}{\tan u+\sin u}}$$

Explain how to use the figure to derive the Pythagorean identities $$\sin ^{2} \theta+\cos ^{2} \theta=1$$ $$1+\tan ^{2} \theta=\sec ^{2} \theta$$ $$\text { and } 1+\cot ^{2} \theta=\csc ^{2} \theta$$ Discuss how to remember these identities and other fundamental trigonometric identities.

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos 3 \beta=\cos ^{3} \beta-3 \sin ^{2} \beta \cos \beta$$

Fill in the blanks. (Note: \(x \rightarrow c^{+}\) indicates that \(x\) approaches \(c\) from the right, and \(x \rightarrow c^{-}\) indicates that \(x\) approaches \(c\) from the left.) $$\text { As } x \rightarrow \frac{\pi}{2}, \sin x \rightarrow$$ ______ $$\text { and } \csc x \rightarrow$$ _____

Determine whether the statement is true or false. Justify your answer. If you correctly solve a trigonometric equation down to the statement \(\sin x=3.4,\) then you can finish solving the equation by using an inverse trigonometric function.

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sin 4 \beta=4 \sin \beta \cos \beta\left(1-2 \sin ^{2} \beta\right)$$

Fill in the blanks. (Note: \(x \rightarrow c^{+}\) indicates that \(x\) approaches \(c\) from the right, and \(x \rightarrow c^{-}\) indicates that \(x\) approaches \(c\) from the left.) $$\text { As } x \rightarrow 0^{+}, \cos x \rightarrow$$ ______ $$\text { and } \sec x \rightarrow$$ ______

Determine whether the statement is true or false. Justify your answer. The equation \(2 \sin 4 t-1=0\) has four times the number of solutions in the interval \([0,2 \pi)\) as the equation \(2 \sin t-1=0\)

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\frac{\sin x \pm \sin y}{\cos x+\cos y}=\tan \frac{x \pm y}{2}$$

Fill in the blanks. (Note: \(x \rightarrow c^{+}\) indicates that \(x\) approaches \(c\) from the right, and \(x \rightarrow c^{-}\) indicates that \(x\) approaches \(c\) from the left.) $$\text { As } x \rightarrow \frac{\pi^{+}}{2}, \tan x \rightarrow$$ _______ $$\text { and cot } x \rightarrow$$ ______

Determine whether the statement is true or false. Justify your answer. Writing Describe the difference between verifying an identity and solving an equation.

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

Fill in the blanks. (Note: \(x \rightarrow c^{+}\) indicates that \(x\) approaches \(c\) from the right, and \(x \rightarrow c^{-}\) indicates that \(x\) approaches \(c\) from the left.) $$\text { As } x \rightarrow \pi^{-}, \sin x \rightarrow$$ _______ $$\text { and csc } x \rightarrow$$ _______.

Write the trigonometric expression as an algebraic expression. $$\sin (2 \arcsin x)$$

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$124^{\circ}$$

Write the trigonometric expression as an algebraic expression. $$\cos (2 \arccos x)$$

Rewrite each trigonometric function of \(\theta\) in terms of \(\cos \theta\)

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$486^{\circ}$$

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

Rewrite the expression in terms of \(\sin \theta\) and \(\cos \theta\) $$\frac{\sec \theta(1+\tan \theta)}{\sec \theta+\csc \theta}$$

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-215.63^{\circ}$$

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

Rewrite the expression in terms of \(\sin \theta\) and \(\cos \theta\) $$\frac{\csc \theta(1+\cot \theta)}{\tan \theta+\cot \theta}$$

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-0.46^{\circ}$$