Chapter 5

Determine whether the statement is true or false. Justify your answer. The graph of \(y=\csc x\) can be obtained on a calculator by graphing the reciprocal of \(y=\sin x\).

Find the reference angle \(\theta^{\prime} .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=-6.5$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\sec \left[\arctan \left(-\frac{3}{5}\right)\right]\)

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=\frac{1}{100} \cos 50 \pi t$$

Rewrite each angle in degree measure. (Do not use a calculator.) (a) \(-\frac{15 \pi}{6}\) (b) \(\frac{28 \pi}{15}\)

True or False Determine whether the statement is true or false. Justify your answer. The tangent function can be used to model harmonic motion.

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\tan \theta=\sqrt{3}\) (b) \(\cos \theta=\frac{\sqrt{2}}{2}\)

Determine whether the statement is true or false. Justify your answer. The graph of \(y=\sec x\) can be obtained on a calculator by graphing a translation of the reciprocal of \(y=\sin x\).

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$225^{\circ}$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\tan \left[\arcsin \left(-\frac{3}{4}\right)\right]\)

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$115^{\circ}$$

True or False Determine whether the statement is true or false. Justify your answer. An example of a bearing used in aviation is \(\mathrm{S} 25^{\circ} \mathrm{W}.\)

Find each value of \(\theta\) in degrees \((0^{\circ}<\theta<90^{\circ})\) and radians \((0 < \theta <\pi / 2)\) without using a calculator. (a) \(\cot \theta=\frac{\sqrt{3}}{3}\) (b) \(\sec \theta=\sqrt{2}\)

Consider the functions $$f(x)=2 \sin x \quad \text { and } \quad g(x)=\frac{1}{2} \csc x$$ on the interval \((0, \pi)\). (a) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (b) Approximate the interval in which \(f(x)>g(x)\). (c) Describe the behavior of each of the functions as \(x\) approaches \(\pi .\) How is the behavior of \(g\) related to the behavior of \(f\) as \(x\) approaches \(\pi ?\)

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$300^{\circ}$$

Find the exact value of the expression. Use a graphing utility to verify your result. (Hint: Make a sketch of a right triangle.) \(\cot \left(\arctan \frac{5}{8}\right)\)

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$83.7^{\circ}$$

True or False Determine whether the statement is true or false. Justify your answer. Writing Find two bearings perpendicular to \(\mathrm{N} 32^{\circ} \mathrm{E}\) and explain how you found them.

Consider the functions given by $$f(x)=\tan \frac{\pi x}{2} \quad \text { and } \quad g(x)=\frac{1}{2} \sec \frac{\pi x}{2}$$ on the interval (-1,1), (a) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (b) Approximate the interval in which \(f(x)

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-750^{\circ}$$

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) cot(arctan \(x\) )

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-216.35^{\circ}$$

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-495^{\circ}$$

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) \(\sin (\arctan x)\)

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-46.52^{\circ}$$

Equation of a Line in Standard Write the standard form of the equation of the line that has the specified characteristics. \(m=4,\) passes through (-1,2)

Use a graphing utility to explore the ratio \(f(x),\) which appears in calculus. (a) Complete the table. Round your results to four decimal places. (b) Use the graphing utility to graph \(f .\) Use the zoom and trace features to describe the behavior of the graph as \(x\) approaches 0 (c) Write a brief statement regarding the value of the ratio based on your results in parts (a) and (b). \(f(x)=\frac{\tan x}{x}\)

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$5 \pi / 3$$

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) \(\sin [\arccos (x+2)]\)

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$-0.78^{\circ}$$

Equation of a Line in Standard Write the standard form of the equation of the line that has the specified characteristics. \(m=-\frac{1}{2},\) passes through \(\left(\frac{1}{3}, 0\right)\)

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$3 \pi / 4$$

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$395^{\circ}$$

Equation of a Line in Standard Write the standard form of the equation of the line that has the specified characteristics. Passes through (-2,6) and (3,2)

Using calculus, it can be shown that the tangent function can be approximated by the polynomial \(\tan x \approx x+\frac{2 x^{3}}{3 !}+\frac{16 x^{5}}{5 !}\) where \(x\) is in radians. Use a graphing utility to graph the tangent function and its polynomial approximation in the same viewing window. How do the graphs compare?

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-5 \pi / 6$$

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) \(\tan \left(\arccos \frac{x}{5}\right)\)

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$\frac{\pi}{7}$$

Equation of a Line in Standard Write the standard form of the equation of the line that has the specified characteristics. Passes through \(\left(\frac{1}{4},-\frac{2}{3}\right)\) and \(\left(-\frac{1}{2}, \frac{1}{3}\right)\)

\((p .409)\) A zip line steel cable is being constructed for a reality television competition show. The high end of the zip line is attached to the top of a 50 -foot pole, while the lower end is anchored at ground level to a stake 50 feet from the base of the pole. (See figure.) (a) Find the angle of elevation of the zip line. (b) Find the number of feet of steel cable needed for the zip line. (c) A contestant takes 6 seconds to reach the ground from the top of the zip line. At what rate is the contestant moving down the line? At what rate is the contestant dropping vertically?

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-4 \pi / 3$$

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) \(\cot \left(\arctan \frac{4}{x}\right)\)

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$\frac{5 \pi}{11}$$

Finding the Domain of a Function Find the domain of the function. $$f(x)=3 x+8$$

You are skiing down a mountain with a vertical height of 1500 feet. The distance from the top of the mountain to the base is 3000 feet. What is the angle of elevation from the base to the top of the mountain?

Identify the rule of algebra illustrated by the statement. \(5(a-9)=5 a-45\)

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) for all real numbers \(x\) in the interval \([-2 \pi, 2 \pi]\) Use the graphs to find the real numbers \(x\) such that \(y_{1}=y_{2}.\) $$\begin{aligned} &y_{1}=\sin x\\\ &y_{2}=-\frac{1}{2} \end{aligned}$$

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$11 \pi / 4$$

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) \(\csc \left(\arctan \frac{x}{\sqrt{7}}\right)\)

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$6.5 \pi$$