Exam 7: The Circular Functions and Their Graphs

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. - $\frac { \pi } { 3 }$ Find the radius of a circle in which a central angle of radian determines a sector of area 57 square meters. Round to the nearest hundredth.

Graph the function over a one-period interval. - $y = 2 + \frac { 1 } { 2 } \cos 4 \left( x + \frac { \pi } { 2 } \right)$

Solve the problem -An object is spinning around a circle with a radius of 24 centimeters. If in 14 seconds a central angle of $\frac { 1 } { 4 }$ radian has been covered, what is the linear speed of the object?

Graph the function over a one-period interval. - $y = - \frac { 1 } { 3 } \cos 4 ( x - \pi )$

Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km. -Find the radius (to the nearest hundredth of a millimeter) of a pulley if rotating the pulley $74.53 ^ { \circ }$ raises the pulley $16.0 \mathrm {~mm}$ .

The function graphed is of the form y $y = a \sin b x \text { or } y = a \cos b x , \text { where } b > 0$ Determine the equation of the graph. -

Find the area of a sector of a circle having radius r and central angle $\theta$ . If necessary, express the answer to the nearest tenth. -Find the measure (in radians) of a central angle of a sector of area 56 square inches in a circle of radius 8 inches. Round to the nearest hundredth.

Graph the function over a one-period interval. -The temperature in Fairbanks is approximated by $T ( x ) = 37 \sin \left[ \frac { 2 \pi } { 365 } ( x - 101 ) \right] + 25$ where $T ( x )$ is the temperature on day $x$ , with $x = 1$ corresponding to Jan. 1 and $x = 365$ correspondin; to Dec. 31. Estimate the temperature on day 31.

Find the exact circular function value. - $\cot \pi$

Find the specified quantity. -Find the period of $y = 5 \sin \left( 6 x + \frac { \pi } { 2 } \right)$ .

Match the function with its graph. -1) $y = \tan x$ 2) $y = \cot x$ 3) $y = - \tan x$ 4) $y = - \cot x$

Find the specified quantity. -Find the amplitude of $y = - 5 \cos \left( 2 x + \frac { \pi } { 4 } \right)$ .

Find the exact circular function value. - $\csc \frac { 5 \pi } { 3 }$

Find the specified quantity. -Find the amplitude of $y = 2 \cos ( 3 x - \pi )$ .

Match the function with its graph. -1) $y = 2 + \sin x$ 2) $y = 2 + \cos x$ 3) $y = - 2 + \sin x$ 4) $y = - 2 + \cos x$

The function graphed is of the form $y = \cos x + c , y = \sin x + c , y = \cos ( x - d ) , \text { or } y = \sin ( x - d )$ where d is the least possible positive value. Determine the equation of the graph. -

Graph the function over a one-period interval. -A coil of wire ro tating in a magnetic field induces a voltage given by $e = 20 \sin \left( \frac { \pi t } { 4 } - \frac { \pi } { 2 } \right)$ where $t$ is time in seconds. Find the smallest positive time to produce a voltage of $10 \sqrt { 2 }$ .

Graph the function. - $y = \cos \frac { 3 } { 2 } x$

The function graphed is of the form y $y = a \sin b x \text { or } y = a \cos b x , \text { where } b > 0$ Determine the equation of the graph. -A weight attached to a spring is pulled down 7 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 5 }$ second, determine a trigonometric model that gives the position of the weight at time $t$ seconds.

Convert the degree measure to radians. Leave answer as a multiple of $\pi .$ - $1080 ^ { \circ }$