Prove that the area of a circular sector of radius r with central angle $\theta$ is

$$A=\frac{1}{2} r^{2} \theta,$$

where $\theta$ is measured in radians.

Convert the angle measure from degrees to radians. Round your answer three decimal places. 225$^{\circ}$

Explain the meanings of (a) an angle in standard position, (b) a negative angle, (c) coterminal angles, and (d) an obtuse angle.

Fill in the blank. The _______ of a sine or cosine curve represents half the distance between the maximum and minimum values of the function.

Find the exact value of each expression, if possible, without using a calculator. (a) $\arccos (-\sqrt{3})$ (b) $\arcsin \frac{\sqrt{2}}{2}$

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. $\theta=4 \pi / 3$ (a) (f + g)($\theta$) (b) (g - f)($\theta$) (c) [g($\theta$)]$^{2}$ (d) (fg)($\theta$) (e) f(2$\theta$) (f) g(-$\theta$)

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. $\theta=-5 \pi / 3$ (a) (f + g)($\theta$) (b) (g - f)($\theta$) (c) [g($\theta$)]$^{2}$ (d) (fg)($\theta$) (e) f(2$\theta$) (f) g(-$\theta$)

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. $\theta=5 \pi / 6$ (a) (f + g)($\theta$) (b) (g - f)($\theta$) (c) [g($\theta$)]$^{2}$ (d) (fg)($\theta$) (e) f(2$\theta$) (f) g(-$\theta$)

Determine whether the statement is true or false. Justify your answer. The angles of a triangle can have radian measures of 2$\pi$/3, $\pi$/4, and $\pi$/12.

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. $\theta=7 \pi / 6$ (a) (f + g)($\theta$) (b) (g - f)($\theta$) (c) [g($\theta$)]$^{2}$ (d) (fg)($\theta$) (e) f(2$\theta$) (f) g(-$\theta$)

Determine whether the statement is true or false. Justify your answer. An angle that measures -$1260^{\circ}$ lies in Quadrant III.

In given exercise, find the exact value of given function for the given angle for $f(\theta)=\sin \theta$ and $g(\theta)=\cos \theta$. Do not use a calculator.

(a) $f(\theta)+g(\theta)$

(b) $g(\theta)-f(\theta)$

(c) $[g(\theta)]^2$

(d) $f(\theta) g(\theta)$

(e) $2 f(\theta)$

(f) $g(-\theta)$

$$\theta=300^{\circ}$$

Determine the period and amplitude.

$$y=\frac{1}{3} \sin 10 x$$

Solve the right triangle shown in the figure. (Round your answers to two decimal places.)

$B=65^{\circ} 12^{\prime}, a=14.2$