5.4 Right Triangle Trigonometry

Find the exact value of the trigonometric functions of $\frac{\pi}{3} ,$ using side lengths.

Find the exact value of the trigonometric functions of $\frac{\pi}{4} ,$ using side lengths.

If $\sin t = \frac{5}{12} ,$ find $\cos \left(\right. \frac{\pi}{2} - t \left.\right) .$

If $\csc \left(\frac{\pi}{6}\right) = 2 ,$ find $\sec \left(\frac{\pi}{3}\right) .$

A right triangle has one angle of $\frac{\pi}{3}$ and a hypotenuse of 20. Find the unknown sides and angle of the triangle.

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of $\frac{5 \pi}{12}$ with the ground? Round to the nearest foot.

The tangent of an angle compares which sides of the right triangle?

Explain the cofunction identity.

Use cofunctions of complementary angles.

$\cos \left(\frac{\pi}{3}\right) = \sin (___)$

Use cofunctions of complementary angles.

$\tan \left(\frac{\pi}{4}\right) = \cot \left(__\right)$

Find the lengths of the missing sides if side $a$ is opposite angle $A ,$ side $b$ is opposite angle $B ,$ and side $c$ is the hypotenuse.

$\sin B = \frac{1}{2} , a = 20$

Find the lengths of the missing sides if side $a$ is opposite angle $A ,$ side $b$ is opposite angle $B ,$ and side $c$ is the hypotenuse.

$\tan A = 100 , b = 100$

Find the lengths of the missing sides if side $a$ is opposite angle $A ,$ side $b$ is opposite angle $B ,$ and side $c$ is the hypotenuse.

$a = 5 , \measuredangle A = 60^{\circ}$

$\sin A$

$\tan A$

$\sec A$

$\sin A$

$\tan A$

$\sec A$

Use a calculator to find the length of each side to four decimal places.

$b = 15 , \measuredangle B = 15^{\circ}$

Use a calculator to find the length of each side to four decimal places.

$c = 50 , \measuredangle B = 21^{\circ}$

Use a calculator to find the length of each side to four decimal places.

$b = 3.5 , \measuredangle A = 78^{\circ}$

A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is $43° ,$ and that the angle of depression to the bottom of the tower is $31° .$ How tall is the tower?

A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is $18° ,$ and that the angle of depression to the bottom of the monument is $3° .$ How far is the person from the monument?

There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be $36° .$ From the same location, the angle of elevation to the top of the lightning rod is measured to be $38° .$ Find the height of the lightning rod.

A 23-ft ladder leans against a building so that the angle between the ground and the ladder is $80° .$ How high does the ladder reach up the side 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.

Convert the angle measures to degrees.

$\frac{\pi}{4}$

Convert the angle measures to radians.

-210°

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°.

Find the angle between 0° and 360° that is coterminal with the given angle.

$420°$

Find the angle between 0 and $2 \pi$ in radians that is coterminal with the given angle.

$- \frac{20 \pi}{11}$

Draw the angle provided in standard position on the Cartesian plane.

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.

Find the exact value of $\sin \frac{\pi}{3} .$

Find the exact value of $\cos \pi .$

State the reference angle for $\frac{3 \pi}{4} .$

Compute sine of $\frac{5 \pi}{4} .$

State the range of the sine and cosine functions.

Find the exact value of the given expression.

$\tan \frac{\pi}{4}$

Find the exact value of the given expression.

$\sec \frac{\pi}{4}$

Use reference angles to evaluate the given expression.

$\sec 315°$

Use reference angles to evaluate the given expression.

If $\text{tan} \left(t\right) = - 0.6 ,$ what is the $\text{tan} \left(- t\right) ?$

Use reference angles to evaluate the given expression.

If $\text{cos} \left(t\right) = \frac{\sqrt{2}}{2} ,$ find $\text{sin} \left(t + 2 \pi\right) .$ There are two possible solutions.

Use reference angles to evaluate the given expression.

Which trigonometric functions are odd?

Use side lengths to evaluate.

$\cot \frac{\pi}{3}$

Use side lengths to evaluate.

$\cos \left(\frac{\pi}{2}\right) = \sin \left(__°\right)$

Use the given information to find the lengths of the other two sides of the right triangle.

$\cos B = \frac{3}{5} , a = 6$

$\sin A$

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 $70° .$ How high does the ladder reach up the side of the building?

Convert $\frac{5 \pi}{6}$ radians to degrees.

Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of $30° .$

Find the angle between $0°$ and $\text{360}°$ that is coterminal with $375° .$

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?

Compute sine of $240° .$

State the range of the sine and cosine functions.

Find the exact value of $\tan \frac{\pi}{3} .$

Use reference angles to evaluate $\tan 210 \circ .$

If $\text{cos} t = \frac{\sqrt{3}}{2} ,$ find $\text{cos} \left(t - 2 \pi\right) .$

Find the missing angle: $\cos \left(\frac{\pi}{6}\right) = \sin \left(___\right)$