Right Triangle Trigonometry Word Problem Examples

Example 1 If B = 42° and a = 12, find c.

From the triangle at the right, you know the measure of an angle and the side adjacent. You want to know the measure of the hypotenuse. The cosine function relates the side adjacent to the angle and the hypotenuse.

a side adjacent cos B = cos = c hypotenuse 12 cos 42° = Substitute 42° for B and 12 for a. c c cos 42° = 12 Multiply each side by c. 12 c = Divide each side by 42°. cos 42° c 16.14759276 Use a calculator.

Therefore, c is about 16.1.

Example 2 RECREATION A child holding on to the string of a kite gets tired and decides to put the string on the ground and secure it with a brick. The length of the string from the brick to the kite is 240 feet. a. If the angle formed by the string and the ground is 50.275°, how high is the kite? b. What is the horizontal distance between the kite and the brick? a. You know the measures of an angle and the hypotenuse. To find the height of the kite, you need to know the measure of the side opposite the given angle. In this case, use the sine function.

h side opposite sin 50.275° = sin = 240 hypotenuse 240 sin 50.275° = h Multiply each side by 240. 184.588984 h Use a calculator.

The kite is about 184.6 feet high.

b. To find the horizontal distance between the kite and the brick, you need to know the length of the side adjacent to the given angle. Use the cosine function.

d side adjacent cos 50.275° = cos = 240 hypotenuse 240 cos 50.275° = d Multiply each side by 240. 153.3848329 d Use a calculator.

The horizontal distance between the kite and the brick is about 153.4 feet.

Example 3 GEOMETRY A regular hexagon is inscribed in a circle with diameter 10.32 centimeters. Find the apothem of the hexagon.

First, draw a diagram. If the diameter of the circle is 10.32 centimeters, the radius is 10.32 2 or 5.16 centimeters. The measure of α is 360° 12 or 30°.

a side adjacent cos 30° = cos = 5.16 hypotenuse 5.16 cos 30° = a Multiply each side by 5.16. 4.468691084 a Use a calculator.

The apothem is about 4.47 centimeters.

Example 4 In Washington, D.C., the Washington Monument is situated between the Capitol and the Lincoln Memorial. A tourist standing at the Lincoln Memorial tilts her head at an angle of 7.491° in order to look up to the top of the Washington Monument. At the same time, another tourist standing at the Capitol steps tilts his head at a 5.463° to also look at the top of the Washington Monument. a. About how high is the Washington Monument? b. What is the distance between the Lincoln Memorial and the Washington monument? a. Draw a diagram to model the situation. Let h represent the height of the Washington Monument in feet. The distance between the Capitol and the Washington Monument is 1.1 miles, which is 5808 feet. Write an equation that relates the distance between the Capitol and the Washington monument and the height of the monument.

h tan 5.463° = 5808 5808 tan 5.463° = h 555.4616096 h Use a calculator.

The Washington Monument is about 555 feet tall. b. Let d represent the distance between the Lincoln Memorial and the Washington Monument. Write an equation that relates this distance and the height of the Washington Monument.

555 tan 7.491° = d d tan 7.491° = 555 555 d = tan 7.491° d 4220.76667

The distance between the Lincoln Memorial and the Washington monument is about 4221 feet, which 4221 is about or 0.8 mile. 5280