Chapter 7

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

$\mathrm{240°}$

$\frac{4\pi }{3}$

$\frac{2\pi }{3}$

$\frac{7\pi }{2}\approx 11.00{\text{in}}^2$

$\frac{81\pi }{20}\approx 12.72{\text{cm}}^2$

$\mathrm{20°}$

$\mathrm{60°}$

$\mathrm{-75°}$

$\frac{\pi }{2}$ radians

$\mathrm{-3}\pi$ radians

$\pi$ radians

$\frac{5\pi }{6}$ radians

$\frac{5.02\pi }{3}\approx 5.26$ miles

$\frac{25\pi }{9}\approx 8.73$ centimeters

$\frac{21\pi }{10}\approx 6.60$ meters

104.7198 cm²

0.7697 in²

$\mathrm{250°}$

$\mathrm{320°}$

$\frac{4\pi }{3}$

$\frac{8\pi }{9}$

$1320$ rad/min $210.085$ RPM

$7$ in./s, 4.77 RPM , $28.65$ deg/s

$1,809,557.37\text{mm/min}=$
$30.16\text{m/s}$

$5.76$ miles

$\mathrm{120°}$

794 miles per hour

2,234 miles per hour

11.5 inches

The tangent of an angle is the ratio of the opposite side to the adjacent side.

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

$\frac{\pi }{6}$

$\frac{\pi }{4}$

$b=\frac{20\sqrt{3}}{3},c=\frac{40\sqrt{3}}{3}$

$a=\mathrm{10,000},c=\mathrm{10,000.5}$

$b=\frac{5\sqrt{3}}{3},c=\frac{10\sqrt{3}}{3}$

$\frac{5\sqrt{29}}{29}$

$\frac{5}{2}$

$\frac{\sqrt{29}}{2}$

$\frac{5\sqrt{41}}{41}$

$\frac{5}{4}$

$\frac{\sqrt{41}}{4}$

$c=14,b=7\sqrt{3}$

$a=15,b=15$

$b=9.9970,c=12.2041$

$a=2.0838,b=11.8177$

$a=55.9808,c=57.9555$

$a=46.6790,b=17.9184$

$a=16.4662,c=16.8341$

188.3159

200.6737

498.3471 ft

1060.09 ft

27.372 ft

22.6506 ft  

368.7633 ft

The unit circle is a circle of radius 1 centered at the origin.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, $t,$ formed by the terminal side of the angle $t$ and the horizontal axis.

The sine values are equal.

I

IV

$\frac{\sqrt{3}}{2}$

$\frac{1}{2}$

$\frac{\sqrt{2}}{2}$

0

-1

$\frac{\sqrt{3}}{2}$

$\mathrm{60°}$

$80°$

$\mathrm{45°}$

$\frac{\pi }{3}$

$\frac{\pi }{3}$

$\frac{\pi }{8}$

$\mathrm{60°},$ Quadrant IV, $\text{sin}(\mathrm{300°})=-\frac{\sqrt{3}}{2}$,$\cos (\mathrm{300°})=\frac{1}{2}$

$\mathrm{45°},$ Quadrant II, $\text{sin}\left(\mathrm{135°}\right)=\frac{\sqrt{2}}{2}$,$\cos (\mathrm{135°})=-\frac{\sqrt{2}}{2}$

$\mathrm{60°},$ Quadrant II, $\text{sin}\left(\mathrm{120°}\right)=\frac{\sqrt{3}}{2}$,$\cos (\mathrm{120°})=-\frac{1}{2}$

$\mathrm{30°},$ Quadrant II, $\text{sin}\left(\mathrm{150°}\right)=\frac{1}{2}$,$\cos (\mathrm{150°})=-\frac{\sqrt{3}}{2}$

$\frac{\pi }{6},$ Quadrant III, $\text{sin}\left(\frac{7\pi }{6}\right)=-\frac{1}{2}$,$\cos \left(\frac{7\pi }{6}\right)=-\frac{\sqrt{3}}{2}$

$\frac{\pi }{4},$ Quadrant II, $\text{sin}\left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}$,$\cos \left(\frac{3\pi }{4}\right)=-\frac{\sqrt[]{2}}{2}$

$\frac{\pi }{3},$ Quadrant II, $\text{sin}\left(\frac{2\pi }{3}\right)=\frac{\sqrt{3}}{2}$,$\cos \left(\frac{2\pi }{3}\right)=-\frac{1}{2}$

$\frac{\pi }{4},$ Quadrant IV, $\text{sin}\left(\frac{7\pi }{4}\right)=-\frac{\sqrt{2}}{2},\text{cos}\left(\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2}$

$\frac{\sqrt{77}}{9}$

$-\frac{\sqrt{15}}{4}$

$\left(\mathrm{-10},10\sqrt{3}\right)$

$\left(\mathrm{–2.778},15.757\right)$

$\left[\mathrm{–1},1\right]$

$\sin t=\frac{1}{2},\cos t=-\frac{\sqrt{3}}{2}$

$\sin t=-\frac{\sqrt{2}}{2},\cos t=-\frac{\sqrt{2}}{2}$

$\sin t=\frac{\sqrt{3}}{2},\cos t=-\frac{1}{2}$

$\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2}$

$\sin t=0,\cos t=-1$

$\sin t=-0.596,\cos t=0.803$

$\sin t=\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}$

$\sin t=-\frac{1}{2},\cos t=\frac{\sqrt{3}}{2}$

$\sin t=0.761,\cos t=-0.649$

$\sin t=1,\cos t=0$

−0.1736

0.9511

−0.7071

−0.1392

−0.7660

$\frac{\sqrt{2}}{4}$

$-\frac{\sqrt{6}}{4}$

$\frac{\sqrt{2}}{4}$

$\frac{\sqrt{2}}{4}$

0

$\left(0,\mathrm{–1}\right)$

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

Yes, when the reference angle is $\frac{\pi }{4}$ and the terminal side of the angle is in quadrants I and III. Thus, a $x=\frac{\pi }{4},\frac{5\pi }{4},$ the sine and cosine values are equal.

Substitute the sine of the angle in for $y$ in the Pythagorean Theorem $x^2+y^2=1.$ Solve for $x$ and take the negative solution.

The outputs of tangent and cotangent will repeat every $\pi$ units.

$\frac{2\sqrt{3}}{3}$

$\sqrt{3}$

$\sqrt{2}$

1

2

$\frac{\sqrt{3}}{3}$

$-\frac{2\sqrt{3}}{3}$

$\sqrt{3}$

$-\sqrt{2}$

–1

-2

$-\frac{\sqrt{3}}{3}$

2

$\frac{\sqrt{3}}{3}$

–2

–1

$\sin t=-\frac{2\sqrt{2}}{3}$, $\sec t=-3$, $\csc t=-\frac{3\sqrt{2}}{4}$, $\tan t=2\sqrt{2}$, $\cot t=\frac{\sqrt{2}}{4}$

$\sec t=2,$ $\csc t=\frac{2\sqrt{3}}{3},$ $\tan t=\sqrt{3},$ $\cot t=\frac{\sqrt{3}}{3}$

$-\frac{\sqrt{2}}{2}$

3.1

1.4

$\sin t=\frac{\sqrt{2}}{2}$, $\cos t=\frac{\sqrt{2}}{2}$, $\tan t=1$, $\cot t=1$, $\sec t=\sqrt{2}$, $\csc t=\sqrt{2}$

$\sin t=-\frac{\sqrt{3}}{2}$, $\cos t=-\frac{1}{2}$$\tan t=\sqrt{3}$, $\cot t=\frac{\sqrt{3}}{3}$, $\sec t=-2$, $\csc t=-\frac{2\sqrt{3}}{3}$

–0.228

–2.414

1.414

1.540

1.556

$\sin \left(t\right)\approx 0.79$

$\csc t\approx 1.16$

even

even

$\frac{\sin t}{\cos t}=\tan t$

13.77 hours, period: $1000\pi$

3.46 inches

$\mathrm{45°}$

$-\frac{7\pi }{6}$

10.385 meters

$\mathrm{60°}$

$\frac{2\pi }{11}$

1036.73 miles per hour

$\frac{\sqrt{2}}{2}$

$\frac{\sqrt{3}}{3}$

$\mathrm{72°}$

$a=\frac{10}{3},c=\frac{2\sqrt{106}}{3}$

$\frac{6}{11}$

$a=\frac{5\sqrt{3}}{2},b=\frac{5}{2}$

369.2136 ft

$\frac{\sqrt{2}}{2}$

$\mathrm{60°}$

$\frac{\sqrt{3}}{2}$

all real numbers

$\frac{\sqrt{3}}{2}$

$\frac{2\sqrt{3}}{3}$

2

–2.5

$\frac{1}{3}$

cosine, secant

$\mathrm{150°}$

6.283 centimeters

$\mathrm{15°}$

3.351 feet per second, $\frac{2\pi }{75}$ radians per second

$a=\frac{9}{2},b=\frac{9\sqrt{3}}{2}$

$\frac{1}{2}$

real numbers

1

$-\sqrt{2}$

–0.68

$\frac{\pi }{3}$