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A 23-ft ladder leans against a building so that the angle between the ground and the ladder is $\text{\hspace{0.17em}}\mathrm{80\xb0}.\text{\hspace{0.17em}}$ How high does the ladder reach up the side of the building?
22.6506 ft
The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.
The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.
368.7633 ft
Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be $\text{\hspace{0.17em}}\mathrm{60\xb0},$ how far from the base of the tree am I?
For the following exercises, convert the angle measures to degrees.
$-\frac{5\pi}{3}$
For the following exercises, convert the angle measures to radians.
Find the length of an arc in a circle of radius 7 meters subtended by the central angle of 85°.
10.385 meters
Find the area of the sector of a circle with diameter 32 feet and an angle of $\text{\hspace{0.17em}}\frac{3\pi}{5}\text{\hspace{0.17em}}$ radians.
For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.
$-\mathrm{80\xb0}$
For the following exercises, find the angle between 0 and $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ in radians that is coterminal with the given angle.
$\frac{14\pi}{5}$
For the following exercises, draw the angle provided in standard position on the Cartesian plane.
$-\frac{\pi}{3}$
Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour.
1036.73 miles per hour
A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car's speed in miles per hour?
Find the exact value of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi}{3}.$
$\frac{\sqrt{3}}{2}$
Find the exact value of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi}{4}.$
Find the exact value of $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\pi .$
–1
State the reference angle for $\text{\hspace{0.17em}}\mathrm{300\xb0}.$
State the reference angle for $\text{\hspace{0.17em}}\frac{3\pi}{4}.$
$\frac{\pi}{4}$
Compute cosine of $\text{\hspace{0.17em}}\mathrm{330\xb0}.$
Compute sine of $\text{\hspace{0.17em}}\frac{5\pi}{4}.$
$-\frac{\sqrt{2}}{2}$
State the domain of the sine and cosine functions.
State the range of the sine and cosine functions.
$\left[\u20131,1\right]$
For the following exercises, find the exact value of the given expression.
$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi}{6}$
$\mathrm{csc}\text{\hspace{0.17em}}\frac{\pi}{3}$
$\mathrm{sec}\text{\hspace{0.17em}}\frac{\pi}{4}$
$\sqrt{2}$
For the following exercises, use reference angles to evaluate the given expression.
$\mathrm{sec}\text{\hspace{0.17em}}\frac{11\pi}{3}$
$\mathrm{sec}\text{\hspace{0.17em}}\mathrm{315\xb0}$
$\sqrt{2}$
If $\text{\hspace{0.17em}}\mathrm{sec}\left(t\right)=-2.5\text{\hspace{0.17em}}$ , what is the $\text{\hspace{0.17em}}\text{sec}(-t)?$
If $\text{\hspace{0.17em}}\text{tan}(t)=-0.6,$ what is the $\text{\hspace{0.17em}}\text{tan}(-t)?$
0.6
If $\text{\hspace{0.17em}}\text{tan}(t)=\frac{1}{3},$ find $\text{\hspace{0.17em}}\text{tan}(t-\pi ).$
If $\text{\hspace{0.17em}}\text{cos}(t)=\frac{\sqrt{2}}{2},$ find $\text{\hspace{0.17em}}\text{sin}(t+2\pi ).$
$\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}-\frac{\sqrt{2}}{2}$
Which trigonometric functions are even?
Which trigonometric functions are odd?
sine, cosecant, tangent, cotangent
For the following exercises, use side lengths to evaluate.
$\mathrm{cos}\text{\hspace{0.17em}}\frac{\pi}{4}$
$\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi}{3}$
$\frac{\sqrt{3}}{3}$
$\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi}{6}$
$\mathrm{cos}\left(\frac{\pi}{2}\right)=\mathrm{sin}(\text{\_\_\xb0})$
0
$\mathrm{csc}(18\text{\xb0})=\mathrm{sec}(\text{\_\_\xb0})$
For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.
$\mathrm{cos}\text{\hspace{0.17em}}B=\frac{3}{5},a=6$
$b=8,c=10$
$\mathrm{tan}\text{\hspace{0.17em}}A=\frac{5}{9},b=6$
For the following exercises, use [link] to evaluate each trigonometric function.
$\mathrm{sin}\text{\hspace{0.17em}}A$
$\frac{11\sqrt{157}}{157}$
$\mathrm{tan}\text{\hspace{0.17em}}B$
For the following exercises, solve for the unknown sides of the given triangle.
A 15-ft ladder leans against a building so that the angle between the ground and the ladder is $\text{\hspace{0.17em}}\mathrm{70\xb0}.\text{\hspace{0.17em}}$ How high does the ladder reach up the side of the building?
14.0954 ft
The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.
Convert $\text{\hspace{0.17em}}\frac{5\pi}{6}\text{\hspace{0.17em}}$ radians to degrees.
$\mathrm{150\xb0}$
Convert $\text{\hspace{0.17em}}\mathrm{-620\xb0}\text{\hspace{0.17em}}$ to radians.
Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of $\text{\hspace{0.17em}}\mathrm{30\xb0}.$
6.283 centimeters
Find the area of the sector with radius of 8 feet and an angle of $\text{\hspace{0.17em}}\frac{5\pi}{4}\text{\hspace{0.17em}}$ radians.
Find the angle between $\text{\hspace{0.17em}}\mathrm{0\xb0}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{360\xb0}\text{\hspace{0.17em}}$ that is coterminal with $\text{\hspace{0.17em}}\mathrm{375\xb0}.$
$\mathrm{15\xb0}$
Find the angle between 0 and $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ in radians that is coterminal with $\text{\hspace{0.17em}}-\frac{4\pi}{7}.$
Draw the angle $\text{\hspace{0.17em}}\mathrm{315\xb0}\text{\hspace{0.17em}}$ in standard position on the Cartesian plane.
Draw the angle $\text{\hspace{0.17em}}-\frac{\pi}{6}\text{\hspace{0.17em}}$ in standard position on the Cartesian plane.
A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?
3.351 feet per second, $\text{\hspace{0.17em}}\frac{2\pi}{75}\text{\hspace{0.17em}}$ radians per second
Find the exact value of $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\frac{\pi}{6}.$
Compute sine of $\text{\hspace{0.17em}}\mathrm{240\xb0}.$
$-\frac{\sqrt{3}}{2}$
State the domain of the sine and cosine functions.
State the range of the sine and cosine functions.
$\left[\u20131,1\right]$
Find the exact value of $\text{\hspace{0.17em}}\mathrm{cot}\text{\hspace{0.17em}}\frac{\pi}{4}.$
Find the exact value of $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\frac{\pi}{3}.$
$\sqrt{3}$
Use reference angles to evaluate $\text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}\frac{7\pi}{4}.$
Use reference angles to evaluate $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}210\xb0.$
$\frac{\sqrt{3}}{3}$
If $\text{\hspace{0.17em}}\text{csc}\text{\hspace{0.17em}}t=0.68,$ what is the $\text{\hspace{0.17em}}\text{csc}(-t)?$
If $\text{\hspace{0.17em}}\text{cos}\text{\hspace{0.17em}}\text{t}=\frac{\sqrt{3}}{2},$ find $\text{\hspace{0.17em}}\text{cos}(t-2\pi ).$
$\frac{\sqrt{3}}{2}$
Which trigonometric functions are even?
Find the missing angle: $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi}{6}\right)=\mathrm{sin}\left(\_\_\_\right)$
$\frac{\pi}{3}$
Find the missing sides of the triangle $\text{\hspace{0.17em}}ABC:\mathrm{sin}\text{\hspace{0.17em}}B=\frac{3}{4},c=12$
Find the missing sides of the triangle.
$a=\frac{9}{2},b=\frac{9\sqrt{3}}{2}$
The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of 2000 feet from the base of the building. Using this information, find the height of the building.
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