Section 1
Review Exercises
$\theta=\frac{15 \pi}{4}$(a) (FIGURE CAN'T COPY)(b) Quadrant IV(c) $\frac{15 \pi}{4}-2 \pi=\frac{7 \pi}{4}$ $\frac{7 \pi}{4}-2 \pi=-\frac{\pi}{4}$
$\theta=-110^{\circ}$(a) (FIGURE CAN'T COPY)(b) Quadrant III(c) Coterminal angles:$$\begin{aligned}& -110^{\circ}+360^{\circ}=250^{\circ} \\& -110^{\circ}-360^{\circ}=-470^{\circ}\end{aligned}$$
$450^{\circ}=450^{\circ}, \frac{\pi \mathrm{rad}}{180^{\circ}}=\frac{5 \pi}{2} \approx 7.854$ radians
$-33^{\circ} 45^{\prime}=-33.75^{\circ}=-33.75^{\circ}, \frac{\pi \mathrm{rad}}{180^{\circ}}$$$=-\frac{3 \pi}{16} \text { radian } \approx-0.589 \text { radian }$$9. $\frac{3 \pi}{10}=\frac{3 \pi}{10} \cdot \frac{180^{\circ}}{\pi \mathrm{rad}}=54.000^{\circ}$11. $-3.5 \mathrm{rad}=-3.5 \mathrm{rad} \cdot \frac{180^{\circ}}{\pi \mathrm{rad}}=-200.535^{\circ}$
$198.4^{\circ}=198^{\circ}+0.4(60)^{\prime}=198^{\circ} 24^{\prime}$
$138^{\circ}=\frac{138 \pi}{180}=\frac{23 \pi}{30}$ radians$$s=r \theta=20\left(\frac{23 \pi}{30}\right) \approx 48.17 \text { inches }$$
$120^{\circ}=\frac{120 \pi}{180}=\frac{2 \pi}{3}$ radians$$A=\frac{1}{2} r^2 \theta=\frac{1}{2}(18)^2\left(\frac{2 \pi}{3}\right) \approx 33929 \text { square inches }$$
$t=\frac{2 \pi}{3}$ corresponds to the point $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$
$t=\frac{7 \pi}{6}$ corresponds to the point$$(x, y)=\left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right)$$
$t=\frac{3 \pi}{4}$ corresponds to the point $(x, y)=\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$$\begin{array}{ll}\sin \frac{3 \pi}{4}=y=\frac{\sqrt{2}}{2} & \csc \frac{3 \pi}{4}=\frac{1}{y}=\sqrt{2} \\\cos \frac{3 \pi}{4}=x=-\frac{\sqrt{2}}{2} & \sec \frac{3 \pi}{4}=\frac{1}{x}=-\sqrt{2} \\\tan \frac{3 \pi}{4}=\frac{y}{x}=-1 & \cot \frac{3 \pi}{4}=\frac{x}{y}=-1\end{array}$$
$\sin \frac{11 \pi}{4}=\sin \frac{3 \pi}{4}=\frac{\sqrt{2}}{2}$
$\sin \left(-\frac{17 \pi}{6}\right)=\sin \left(-\frac{5 \pi}{6}\right)=-\frac{1}{2}$
$\tan 33 \approx-75.3130$
$\sec \left(\frac{12 \pi}{5}\right)=\frac{1}{\cos \left(\frac{12 \pi}{5}\right)} \approx 3.2361$
opp $=4$, adj $=5$, hyp $=\sqrt{4^2+5^2}=\sqrt{41}$$$\begin{array}{ll}\sin \theta=\frac{\text { opp }}{\text { hyp }}=\frac{4}{\sqrt{41}}=\frac{4 \sqrt{41}}{41} & \csc \theta=\frac{\text { hyp }}{\text { opp }}=\frac{\sqrt{41}}{4} \\\cos \theta=\frac{\text { adj }}{\text { hyp }}=\frac{5}{\sqrt{41}}=\frac{5 \sqrt{41}}{41} & \sec \theta=\frac{\text { hyp }}{\text { adj }}=\frac{\sqrt{41}}{5} \\\tan \theta=\frac{\text { opp }}{\text { adj }}=\frac{4}{5} & \cos \theta=\frac{\text { adj }}{\text { opp }}=\frac{5}{4}\end{array}$$
$\tan 33^{\circ}=0.6494$
$$\text { } \begin{aligned}\cot 15^{\circ} 14^{\prime} & =\frac{1}{\tan \left(15+\frac{14}{60}\right)} \\& \approx 3.6722\end{aligned}$$
$\sin \theta=\frac{1}{3}$(a) $\csc \theta=\frac{1}{\sin \theta}=3$(c) $\sec \theta=\frac{1}{\cos \theta}=\frac{3}{2 \sqrt{2}}=\frac{3 \sqrt{2}}{4}$(b) $\sin ^2 \theta+\cos ^2 \theta=1$(d) $\tan \theta=\frac{\sin \theta}{\cos \theta}=\frac{1 / 3}{(2 \sqrt{2}) / 3}=\frac{1}{2 \sqrt{2}}=\frac{\sqrt{2}}{4}$
$$\begin{aligned}& \sin 1^{\circ} 10^{\prime}=\frac{\pi}{3.5} \\& x=3.5 \sin 1^{\circ} 10^{\prime} \approx 0.07 \text { kilometer or } 71.3 \text { meters } \\&\end{aligned}$$(FIGURE CAN'T COPY)Niralown be male
$x=12, y=16, r=\sqrt{144+256}=\sqrt{400}=20$$x=3.5 \sin 1^{\circ} 10^{\prime}=0.07$ kilometer or 71.3 meters$\sin \theta=\frac{y}{r}=\frac{4}{5}$$\csc \theta=\frac{r}{y}=\frac{5}{4}$$\cos \theta=\frac{x}{r}=\frac{3}{5}$$\sec \theta=\frac{r}{x}=\frac{5}{3}$$\tan \theta=\frac{y}{x}=\frac{4}{3} \quad \cot \theta=\frac{x}{y}=\frac{3}{4}$
$$\begin{array}{ll}x=0.3, y=0.4 \\r=\sqrt{(0.3)^2+(0.4)^2}=0.5 & \csc \theta=\frac{r}{y}=\frac{0.5}{0.4}=\frac{5}{4}=1.25 \\\sin \theta=\frac{y}{r}=\frac{0.4}{0.5}=\frac{4}{5}=0.8 & \sec \theta=\frac{r}{x}=\frac{0.5}{0.3}=\frac{5}{3}=1.67 \\\cos \theta=\frac{x}{r}=\frac{0.3}{0.5}=\frac{3}{5}=0.6 & \cot \theta=\frac{x}{y}=\frac{0.3}{0.4}=\frac{3}{4}=0.75 \\\tan \theta=\frac{y}{x}=\frac{0.4}{0.3}=\frac{4}{3}=1.33 &\end{array}$$
$\sec \theta=\frac{6}{5}, \tan \theta<0 \Rightarrow \theta$ is in Quadrant IV.$$\begin{aligned}& r=6, x=5, y=-\sqrt{36-25}=-\sqrt{11} \\& \sin \theta=\frac{y}{r}=-\frac{\sqrt{11}}{6} \\& \cos \theta=\frac{x}{r}=\frac{5}{6} \\& \tan \theta=\frac{y}{x}=-\frac{\sqrt{11}}{5} \\& \csc \theta=\frac{r}{y}=-\frac{6 \sqrt{11}}{11} \\& \sec \theta=\frac{6}{5} \\& \cot \theta=-\frac{5 \sqrt{11}}{11}\end{aligned}$$
$$\begin{aligned}& \cos \theta=\frac{x}{r}=\frac{-2}{5} \Rightarrow y^2=21 \\& \sin \theta>0 \Rightarrow \theta \text { is in Quadrant II } \Rightarrow y=\sqrt{21} \\& \sin \theta=\frac{y}{r}=\frac{\sqrt{21}}{5} \\& \tan \theta=\frac{y}{x}=-\frac{\sqrt{21}}{2} \\& \csc \theta=\frac{r}{y}=\frac{5}{\sqrt{21}}=\frac{5 \sqrt{21}}{21} \\& \sec \theta=\frac{r}{x}=\frac{5}{-2}=-\frac{5}{2} \\& \cot \theta=\frac{x}{y}=\frac{-2}{\sqrt{21}}=-\frac{2 \sqrt{21}}{21}\end{aligned}$$
$$\begin{aligned}& \theta=264^{\circ} \\& \theta=264^{\circ}-180^{\circ}=84^{\circ}\end{aligned}$$(FIGURE CAN'T COPY)
$$\begin{aligned}\theta & =-\frac{6 \pi}{5} \\-\frac{6 \pi}{5}+2 \pi & =\frac{4 \pi}{5} \\\theta^{\prime} & =\pi-\frac{4 \pi}{5}=\frac{\pi}{5}\end{aligned}$$(FIGURE CAN'T COPY)
$\sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}$$\cos \frac{\pi}{3}=\frac{1}{2}$$\tan \frac{\pi}{3}=\sqrt{3}$
$\sin 495^{\circ}=\sin 45^{\circ}=\frac{\sqrt{2}}{2}$$\cos 495^{\circ}=-\cos 45^{\circ}=-\frac{\sqrt{2}}{2}$$\tan 495^{\circ}=-\tan 45^{\circ}=-1$
$\sin 4 \approx-0.7568$
$\sin \frac{12 \pi}{5} \approx 0.9511$
$y=\sin 6 x$Amplitude: 1Period: $\frac{2 \pi}{6}=\frac{\pi}{3}$(FIGURE CAN'T COPY)
$y=5+\sin x$Amplitude: 1Period: $2 \pi$Shift the graphof $y=\sin x$5 units upward(FIGURE CAN'T COPY)
$g(t)=\frac{5}{2} \sin (t-\pi)$Amplitude: $\frac{5}{2}$Period: $2 \pi$(FIGURE CAN'T COPY)
$y=a \sin b x$(a) $a=2$,$$\begin{aligned}& \frac{2 \pi}{b}=\frac{1}{264} \Rightarrow b=528 \pi \\& y=2 \sin 528 \pi x\end{aligned}$$(b) $f=\frac{1}{1 / 264}$$=264$ cycles per scoond
$f(t)=\tan \left(t+\frac{\pi}{2}\right)$(FIGURE CAN'T COPY)
$f(x)=\frac{1}{2} \csc \frac{x}{2}$(FIGURE CAN'T COPY)
$f(x)=x \cos x$Damping factor: $x$As $x \rightarrow+\infty, f(x)$ oscillates.(FIGURE CAN'T COPY)
$\arcsin (-1)=-\frac{\pi}{2}$
$\operatorname{arccot} \sqrt{3}=\frac{\pi}{6}$
$\tan ^{-1}(-1.5) \approx-0.98$ radian
$\operatorname{arccot}(10.5)=\arctan \left(\frac{1}{10.5}\right)=0.09$
$f(x)=\arctan \left(\frac{x}{2}\right)=\tan ^{-1}\left(\frac{x}{2}\right)$(FIGURE CAN'T COPY)
Let $u=\arctan \frac{3}{4}$ then $\tan u=\frac{3}{4}$.$$\cos \left(\arctan \frac{3}{4}\right)=\frac{4}{5}$$(FIGURE CAN'T COPY)
$\sec \left[\sin ^{-1}\left(-\frac{1}{4}\right)\right]$Let $y=\sin ^{-1}\left(-\frac{1}{4}\right)$ then $\sin y=-\frac{1}{4}$ and$$\sec \left[\sin ^{-2}\left(-\frac{1}{4}\right)\right]=\sec y=\frac{4 \sqrt{15}}{15} \text {. }$$(FIGURE CAN'T COPY)
Let $y=\arccos \left(\frac{x}{2}\right)$ Then$\cos y=\frac{x}{2}$ and $\tan y=\tan \left(\arccos \left(\frac{x}{2}\right)\right)=\frac{\sqrt{4-x^2}}{x}$(FIGURE CAN'T COPY)
$\tan \theta=\frac{70}{30}$$$\theta=\arctan \left(\frac{20}{30}\right) \approx 66.8^{\circ}$$(FIGURE CAN'T COPY)
$$\text {} \left.\begin{array}{rl}\sin 48^{\circ} & =\frac{d_1}{650} \Rightarrow d_1 \approx 483 \\\cos 25^{\circ} & =\frac{d_2}{810} \Rightarrow d_2=734\end{array}\right\} d_1+d_2=1217$$(FIGURE CAN'T COPY)The distance is 1221 miles and the bearing is $85.6^{\circ}$,
False. For each $\theta$ there corresponds exactly one value of $y$.
$f(\theta)=\sec \theta$ is undefined at the zeros of $g(\theta)=\cos \theta$ because $\sec \theta=\frac{1}{\cos \theta}$
The ranges for the other four trigonometric functions are not bounded. For $y=\tan x$ and $y=\cot x$, the range is $(-\infty, \infty)$. For $y=\sec x$ and $y=\csc x$, the range is $(-\infty,-1] \cup[1, \infty)$
Answers will vary.