Chapter 9

The formula \(\omega=\frac{\theta}{t}\) can be rewritten as \(\theta=\) wt. Substituting wt for \(\theta\) changes \(s=r \theta\) to \(s=r \omega t\). Use the formula \(s=r \omega t\) to find the value of the missing variable. \(r=9\) yards, \(\omega=\frac{2 \pi}{5}\) radians per second, \(t=12\) seconds

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) function value. Do not use a calculator. $$\cos \theta=\frac{\sqrt{2}}{2}$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\sin \theta=\frac{1}{2}, \csc \theta=2$$

The formula \(\omega=\frac{\theta}{t}\) can be rewritten as \(\theta=\) wt. Substituting wt for \(\theta\) changes \(s=r \theta\) to \(s=r \omega t\). Use the formula \(s=r \omega t\) to find the value of the missing variable. \(s=\frac{3 \pi}{4}\) kilometers, \(r=2\) kilometers, \(t=4\) seconds

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) and has the given function value. Give calculator approximations to as many decimal places as your calculator displays. $$\cos \theta=0.68716510$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\tan \theta=2, \cot \theta=-2$$

The formula \(\omega=\frac{\theta}{t}\) can be rewritten as \(\theta=\) wt. Substituting wt for \(\theta\) changes \(s=r \theta\) to \(s=r \omega t\). Use the formula \(s=r \omega t\) to find the value of the missing variable. \(s=\frac{3 \pi}{4}\) kilometers, \(r=2\) kilometers, \(t=4\) seconds

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) and has the given function value. Give calculator approximations to as many decimal places as your calculator displays. $$\cos \theta \approx 0.96476120$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. $$\cos \theta=-2, \sec \theta=\frac{1}{2}$$

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) and has the given function value. Give calculator approximations to as many decimal places as your calculator displays. $$\sin \theta=0.41298643$$

Decide whether each statement is possible for some angle \(\theta\), or impossible for that angle. CONCEPT CHECK Is there an angle \(\theta\) for which \(\tan \theta\) and \(\cot \theta\) are both undefined?

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) and has the given function value. Give calculator approximations to as many decimal places as your calculator displays. $$\sin \theta=0.63898531$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\tan \theta=-\frac{15}{8},\) given that \(\theta\) is in quadrant II

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) and has the given function value. Give calculator approximations to as many decimal places as your calculator displays. $$\tan \theta \approx 0.87692035$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\cos \theta=-\frac{3}{5},\) given that \(\theta\) is in quadrant III

Find all values of \(\theta\) if \(\theta\) is in the interval \(\left[0^{\circ}, 360^{\circ}\right)\) and has the given function value. Give calculator approximations to as many decimal places as your calculator displays. $$\tan \theta=1.2841996$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\sin \theta=\frac{\sqrt{5}}{7},\) given that \(\theta\) is in quadrant \(I\)

Find no angles in the interval \([0,2 \pi)\) that satisfy the given equation. Give calculator approximations to as many digits as your calculator displays. $$\tan \theta=0.21264138$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\tan \theta=\sqrt{3},\) given that \(\theta\) is in quadrant III

The speedometer of a pickup truck is designed to be accurate with tires of radius 14 inches. (a) Find the number of rotations, to the nearest whole number, of a tire in 1 hour if the truck is driven \(55 \mathrm{mph}\) (b) Suppose that oversize tires of radius 16 inches are placed on the truck. If the truck is now driven for 1 hour with the speedometer reading 55 mph, how far has the truck gone? If the speed limit is \(60 \mathrm{mph}\) did the driver exceed the speed limit?

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\cot \theta=\frac{\sqrt{3}}{8},\) given that \(\theta\) is in quadrant I

Approximate the area of a sector of a circle having radius \(r\) and central angle \(\boldsymbol{\theta}.\) $$r=29.2 \text { meters; } \theta=\frac{5 \pi}{6} \text { radians }$$

Find no angles in the interval \([0,2 \pi)\) that satisfy the given equation. Give calculator approximations to as many digits as your calculator displays. $$\cot \theta \approx 0.29949853$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\csc \theta=2,\) given that \(\theta\) is in quadrant II

Approximate the area of a sector of a circle having radius \(r\) and central angle \(\boldsymbol{\theta}.\) \(r=59.8\) kilometers \(; \theta=\frac{2 \pi}{3}\) radians

Find no angles in the interval \([0,2 \pi)\) that satisfy the given equation. Give calculator approximations to as many digits as your calculator displays. $$\csc \theta=1.0219553$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). $$\sin \theta=\frac{\sqrt{2}}{6}, \text { given that } \cos \theta<0$$

Approximate the area of a sector of a circle having radius \(r\) and central angle \(\boldsymbol{\theta}.\) \(r=12.7\) centimeters; \(\theta=81.0^{\circ}\)

In a square window of your calculator that gives a good picture of the first quadrant, graph the line \(y=\sqrt{3} x\) with \(x \geq 0 .\) Then, trace to any point on the line. See the figure. What we see is the terminal side of a standard position angle in quadrant I. Store the values of \(\mathrm{X}\) and \(\mathrm{Y}\) in convenient memory locations, and call them \(\mathrm{X}_{1}\) and \(\mathrm{Y}_{1}\). Calculate the value of \(\sqrt{\mathrm{X}_{1}^{2}+\mathrm{Y}_{1}^{2}}\) and store it in a convenient memory location. (Call it \(r\) ) What does this number mean geometrically?

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). $$\cos \theta=\frac{\sqrt{5}}{8}, \text { given that } \tan \theta<0$$

Approximate the area of a sector of a circle having radius \(r\) and central angle \(\boldsymbol{\theta}.\) $$r=18.3 \text { meters; } \theta=125^{\circ}$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). $$\sec \theta=-4, \text { given that } \sin \theta>0$$

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). $$\csc \theta=-3, \text { given that } \cos \theta>0$$

Use fundamental identities to find each expression. Write \(\cos \theta\) in terms of \(\sin \theta\) if \(\theta\) is acute.

Use fundamental identities to find each expression. $$\text { Write } \sec \theta \text { in terms of } \cos \theta$$

A 300 -megawatt solar-power plant needs approximately \(950,000\) square meters of land area to collect the required amount of energy from sunlight. (a) If the land area is circular, approximate its radius. (b) If the land area is a \(35^{\circ}\) sector of a circle, approximate its radius.

In a square window of your calculator that gives a good picture of the first quadrant, graph the line \(y=\sqrt{3} x\) with \(x \geq 0 .\) Then, trace to any point on the line. See the figure. What we see is the terminal side of a standard position angle in quadrant I. Store the values of \(\mathrm{X}\) and \(\mathrm{Y}\) in convenient memory locations, and call them \(\mathrm{X}_{1}\) and \(\mathrm{Y}_{1}\). Look at the equation of the line you graphed, and make a conjecture: The _______ of a line passing through the origin is equal to the _______ of the angle it forms with the positive \(x\) -axis.

Use fundamental identities to find each expression. Write \(\sin \theta\) in terms of \(\cot \theta\) if \(\theta\) is in quadrant III.

Use fundamental identities to find each expression. Write \(\tan \theta\) in terms of \(\cos \theta\) if \(\theta\) is in quadrant IV.

Use fundamental identities to find each expression. Write \(\tan \theta\) in terms of \(\sin \theta\) if \(\theta\) is in quadrant I or IV.

The unusual corral in the figure is separated into 26 areas, many of which approximate sectors of a circle. Assume that the corral has a diameter of 50 meters. (a) Approximate the central angle for each region, assuming that the 26 regions are all equal sectors with the fences meeting at the center. (b) What is the area of each sector (to the nearest square meter)?

Use fundamental identities to find each expression. Write \(\sin \theta\) in terms of \(\sec \theta\) if \(\theta\) is in quadrant I or II.

Work each problem. Derive the identity \(1+\cot ^{2} \theta=\csc ^{2} \theta\) by dividing each side of the equation \(x^{2}+y^{2}=r^{2}\) by \(y^{2}\)

A bicycle has a tire 26 inches in diameter that is rotating at 15 radians per second. Approximate the speed of the bicycle in feet per second and in miles per hour.

When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light and the direction in which the ray is traveling change. (This is why a fish under water is in a different position from where it appears to be.) The changes are given by Snell's law, $$\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$$ where \(c_{1}\) is the speed of light in the first medium, \(c_{2}\) is the speed of light in the second medium, and \(\theta_{1}\) and \(\theta_{2}\) are the angles shown in the figure below. In Exercises assume that \(c_{1}=3 \times 10^{8}\) meters per second. (Figure cant copy) Approximate the speed of light in the second medium. $$\theta_{1}=46^{\circ} ; \theta_{2}=31^{\circ}$$

Work each problem. Using a method similar to the one given in this section showing that \(\frac{\sin \theta}{\cos \theta}=\tan \theta,\) show that \(\frac{\cos \theta}{\sin \theta}=\cot \theta\)

When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light and the direction in which the ray is traveling change. (This is why a fish under water is in a different position from where it appears to be.) The changes are given by Snell's law, $$\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$$ where \(c_{1}\) is the speed of light in the first medium, \(c_{2}\) is the speed of light in the second medium, and \(\theta_{1}\) and \(\theta_{2}\) are the angles shown in the figure below. In Exercises assume that \(c_{1}=3 \times 10^{8}\) meters per second. (Figure cant copy) Approximate the speed of light in the second medium. $$\theta_{1}=39^{\circ} ; \theta_{2}=28^{\circ}$$

Work each problem. CONCEPT CHECK True or false? For all angles \(\theta\), \(\sin \theta+\cos \theta=1 .\) If false, give an example showing why.

The wheels on a skateboard have diameter 2.25 inches. If a skateboarder is traveling downhill at \(15.0 \mathrm{mph}\), approximate the angular speed of the wheels in radians per second.

When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light and the direction in which the ray is traveling change. (This is why a fish under water is in a different position from where it appears to be.) The changes are given by Snell's law, $$\frac{c_{1}}{c_{2}}=\frac{\sin \theta_{1}}{\sin \theta_{2}}$$ where \(c_{1}\) is the speed of light in the first medium, \(c_{2}\) is the speed of light in the second medium, and \(\theta_{1}\) and \(\theta_{2}\) are the angles shown in the figure below. In Exercises assume that \(c_{1}=3 \times 10^{8}\) meters per second. (Figure cant copy) Approximate the speed of light in the second medium. Find \(\theta_{2}\) for the following values of \(\theta_{1}\) and \(c_{2} .\) Round to the nearest degree. \(\theta_{1}=40^{\circ} ; c_{2}=1.5 \times 10^{8}\) meters per second