Chapter 9

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ},(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ}\), and so on. Decide whether each expression is equal to \(0,1\), or \(-1\) or is undefined. $$\cos \left[(2 n+1) \cdot 180^{\circ}\right]$$

Use the identity \(\cos ^{2} s+\sin ^{2} s=1\) to find the value of \(x\) or \(y,\) as appropriate. Then, assuming that \(s\) corresponds to the given point on the unit circle, find the six circular function values for \(s\). $$\left(-\frac{1}{3}, y\right), y<0$$

Convert each radian measure to degrees. Round answers to the nearest minute. $$2$$

Use a calculator to find a decimal approximation for each value. Give as many digits as your calculator displays. $$\tan 5$$

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ},(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ}\), and so on. Decide whether each expression is equal to \(0,1\), or \(-1\) or is undefined. $$\cos \left(n \cdot 360^{\circ}\right)$$

Use the identity \(\cos ^{2} s+\sin ^{2} s=1\) to find the value of \(x\) or \(y,\) as appropriate. Then, assuming that \(s\) corresponds to the given point on the unit circle, find the six circular function values for \(s\). $$\left(-\frac{1}{4}, y\right), y<0$$

Convert each radian measure to degrees. Round answers to the nearest minute. $$5$$

Use a calculator to find a decimal approximation for each value. Give as many digits as your calculator displays. $$\sec 10$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\sin \theta>0, \csc \theta>0$$

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=\frac{1}{2}, \cos s=\frac{\sqrt{3}}{2}$$

Convert each radian measure to degrees. Round answers to the nearest minute. $$1.74$$

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=120^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\cos \theta>0, \sec \theta>0$$

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=\frac{3}{4}, \cos s=\frac{\sqrt{7}}{4}$$

Convert each radian measure to degrees. Round answers to the nearest minute. $$0.3417$$

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=135^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\cos \theta>0, \sin \theta>0$$

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=\frac{4}{5}, \cos s=-\frac{3}{5}$$

Convert each radian measure to degrees. Round answers to the nearest minute. $$-1.3$$

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=240^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\sin \theta>0, \tan \theta>0$$

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=-\frac{1}{2}, \cos s=-\frac{\sqrt{3}}{2}$$

Convert each radian measure to degrees. Round answers to the nearest minute. $$-4$$

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=315^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\tan \theta<0, \cos \theta<0$$

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=-\frac{\sqrt{3}}{2}, \cos s=\frac{1}{2}$$

The monthly average temperatures in degrees Fahrenheit at Mould Bay, Canada, may be modeled by \(f(x)=34 \sin \left[\frac{\pi}{6}(x-4.3)\right],\) where \(x\) is the month and \(x=1\) corresponds to January. (Source: A. Miller and J. Thompson, Elements of Meteorology, Charles E. Merrill.) (a) Find the amplitude, period, and phase shift. (b) Approximate the average temperature during May and December. (c) Estimate the yearly average temperature at Mould Bay.

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=-120^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\cos \theta<0, \sin \theta<0$$

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=\frac{12}{13}, \cos s=\frac{5}{13}$$

The monthly average temperatures in degrees Fahrenheit at Austin, Texas, are given by \(f(x)=17.5 \sin \left[\frac{\pi}{6}(x-4)\right]+67.5,\) where \(x\) is the month and \(x=1\) corresponds to January. (Source: A. Miller and J. Thompson.) (a) Find the amplitude, period, phase shift, and vertical shift. (b) Determine the maximum and minimum monthly average temperatures and the months when they occur. (c) Make a conjecture as to how the yearly average temperature might be related to \(f(x)\)

The term grade has several different meanings in construction work. Some engineers use the term to represent \(\frac{1}{100}\) of a right angle and express it as a percent. For instance, an angle of \(0.9^{\circ}\) would be referred to as a \(1 \%\) grade. (Source: Hay, W., Railroad Engineering. John Wiley and Sons.) (a) By what number should you multiply a grade to convert it to radians? (b) In a rapid-transit rail system, the maximum grade allowed between two stations is \(3.5 \% .\) Express this angle in degrees and in radians.

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=-45^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\sec \theta>0, \csc \theta>0$$

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\sin s>0, \cos s<0$$

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=-315^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\csc \theta>0, \cot \theta>0$$

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\cos s>0, \tan s>0$$

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=-210^{\circ}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\sec \theta<0, \csc \theta<0$$

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\sec s<0, \csc s<0$$

The monthly average temperatures in a Canadian city are shown in the table. $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Month } & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline \text { Temperature ('F) } & 40 & 43 & 47 & 52 & 59 & 63\end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Month } & 7 & 8 & 9 & 10 & 11 & 12 \\\\\hline \text { Temperature }\left(^{\circ} \mathrm{F}\right) & 68 & 67 & 61 & 54 & 47 & 43\end{array}$$ (a) Plot the average monthly temperature over a 24 -month period by letting \(x=1\) and \(x=13\) correspond to January. (b) Find the constants \(a, b, c,\) and \(d\) so that the function \(f(x)=a \sin [(b(x-c)]+d\) models the data. (c) Graph \(f\) together with the data.

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\sin \frac{7 \pi}{6}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\cot \theta<0, \sec \theta<0$$

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\tan s>0, \cos s<0$$

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\cos \frac{5 \pi}{3}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\sin \theta<0, \csc \theta<0$$

Decide in what quadrant the point corresponding to s must lie to satisfy the following conditions for s. $$\cos s>0, \sin s<0$$

For each expression, (a) write the function in terms of a function of the reference angle. (b) give the exact value, and (c) use a calculator to show that the decimal value or approximation for the given function is the same as the decimal value or approximation for your answer in part (b). $$\tan \frac{3 \pi}{4}$$

Identify the quadrant (or possible quadrants) of an angle \(\theta\) that satisfies the given conditions. $$\tan \theta<0, \cot \theta<0$$