Chapter 5

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=315^{\circ}$$

The formulas for the area of a circular sector and arc length are \(A=\frac{1}{2} r^{2} \theta\) and \(s=r \theta,\) respectively. \((r \text { is the radius and } \theta\) is the angle measured in radians.) (a) Let \(\theta=0.8 .\) Write the area and arc length as functions of \(r .\) What is the domain of each function? Use a graphing utility to graph the functions. Use the graphs to determine which function changes more rapidly as \(r\) increases. Explain. (b) Let \(r=10\) centimeters. Write the area and arc length as functions of \(\theta .\) What is the domain of each function? Use a graphing utility to graph and identify the functions.

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\tan \theta=2\)

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=225^{\circ}$$

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) $$\cos \theta=\frac{3}{4}$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-150^{\circ}$$

In your own words, write a definition of 1 radian.

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\sec \theta=3\)

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-300^{\circ}$$

In your own words, explain the difference between 1 radian and 1 degree.

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=7 \pi / 6$$

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=(x-1)^{3}$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=5 \pi / 6$$

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=x^{3}-4$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=4 \pi / 3$$

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=2-x^{3}$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-5 \pi / 3$$

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=-(x+3)^{3}$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-270^{\circ}$$

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=(x+1)^{3}-3$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=180^{\circ}$$

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) $$g(x)=(x-5)^{3}+1$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=7 \pi / 2$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=5 \pi / 2$$

The normal daily high temperature \(T\) (in degrees Fahrenheit) in Savannah, Georgia, can be approximated by $$T=76.4+16 \cos \left(\frac{\pi t}{6}-\frac{7 \pi}{6}\right)$$ where \(t\) is the time (in months), with \(t=1\) corresponding to January. Find the normal daily high temperature for each month. (Source: National Climatic Data Center) (a) January (b) July (c) October

A company that produces wakeboards forecasts monthly sales \(S\) during a two- year period to be $$S=2.7+0.142 t+2.2 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right)$$ where \(S\) is measured in hundreds of units and \(t\) is the time (in months), with \(t=1\) corresponding to January \(2014 .\) Estimate sales for each month. (a) January 2014 (b) February 2015 (c) May 2014 (d) June 2015

Aeronautics An airplane flying at an altitude of 6 miles is on a flight path that passes directly over an observer (see figure). Let \(\theta\) be the angle of elevation from the observer to the plane. Find the distance from the observer to the plane when (a) \(\theta=30^{\circ}\) (b) \(\theta=90^{\circ},\) and \((\mathrm{c}) \theta=120^{\circ}\).

The displacement from equilibrium of an oscillating weight suspended by a spring is given by $$y(t)=\frac{1}{4} \cos 6 t$$ where \(y\) is the displacement (in feet) and \(t\) is the time (in seconds) (see figure). Find the displacement when (a) \(t=0,(b) t=\frac{1}{4},\) and \((c) t=\frac{1}{2}\)

Determine whether the statement is true or false. Justify your answer. $$\sin \theta<\tan \theta \text { in Quadrant I }$$

Determine whether the statement is true or false. Justify your answer. $$\sin \theta < \cos \theta \text { for } 0^{\circ} < \theta < 45^{\circ}$$

Determine whether the statement is true or false. Justify your answer. $$\sin \theta=-\sqrt{1-\cos ^{2} \theta} \text { for } 90^{\circ} < \theta < 180^{\circ}$$

Determine whether the statement is true or false. Justify your answer. $$\cos \theta=-\sqrt{1-\sin ^{2} \theta} \text { for } 90^{\circ} < \theta < 180^{\circ}$$

(a) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|} \hline \theta & 0^{\circ} & 20^{\circ} & 40^{\circ} & 60^{\circ} & 80^{\circ} \\\ \hline \sin \theta & & & & & \\ \hline \sin \left(180^{\circ}-\theta\right) & & & & & \\ \hline \end{array}$$ (b) Make a conjecture about the relationship between \(\sin \theta\) and \(\sin \left(180^{\circ}-\theta\right)\)

Use the procedure in Exercise 143 and a graphing utility to create a table of values and make a conjecture about the relationship between \(\cos \theta\) and \(\cos \left(180^{\circ}-\theta\right)\) for an acute angle \(\theta\).

(a) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 0 & 0.3 & 0.6 & 0.9 & 1.2 & 1.5 \\ \hline \cos \left(\frac{3 \pi}{2}-\theta\right) & & & & & & \\ \hline-\sin \theta & & & & & & \\ \hline \end{array}$$ (b) Make a conjecture about the relationship between \(\cos \left(\frac{3 \pi}{2}-\theta\right)\) and \(-\sin \theta\).

Use a graphing utility to create a table of values to compare tan \(t\) with \(\tan (t+2 \pi), \tan (t+\pi)\) and \(\tan (t+\pi / 2)\) for \(t=0,0.3,0.6,0.9,1.2,\) and 1.5 Use your results to make a conjecture about the period of the tangent function. Explain your reasoning.

Because \(f(t)=\sin t\) is an odd function and \(g(t)=\cos t\) is an even function, what can be said about the function \(h(t)=f(t) g(t) ?\)

Your classmate uses a calculator to evaluate \(\tan (\pi / 2)\) and gets a result of 0.0274224385 Describe the error.

Write a study sheet that will help you remember how to evaluate the six trigonometric functions of any angle \(\theta\) in standard position. Include figures and diagrams as needed.

Solve the equation. Round your answer to three decimal places, if necessary. $$3 x-7=14$$

Solve the equation. Round your answer to three decimal places, if necessary. $$44-9 x=61$$

Solve the equation. Round your answer to three decimal places, if necessary. $$x^{2}-2 x-5=0$$

Solve the equation. Round your answer to three decimal places, if necessary. $$2 x^{2}+x-4=0$$

Solve the equation. Round your answer to three decimal places, if necessary. $$\frac{3}{x-1}=\frac{x+2}{9}$$

Solve the equation. Round your answer to three decimal places, if necessary. $$\frac{5}{x}=\frac{x+4}{2 x}$$