Application problems

Line of sight Angle of Elevation

Angle of Depression Line of sight

Technique Draw a picture – always draw triangle same direction Put the known information on the triangle. Use trig to find missing information

line of sight, ladder, ramp Angle with wall Tall, Height, Altitude, Above   line of sight, ladder, ramp Angle of Elevation Angle of Depression Angle with ground Ground, base, horizontal distance, shadow, away, across

Example 1 Suppose you have been assigned the job of measuring the height of the local radio tower. Climbing makes you dizzy, so you decide to do the whole job at ground level. From a point of 47.3 meters from the base of the tower, you find that you must look up at an angle of 53º to see the top of the tower. How high is the tower? 53º 47.3

Example 2: Fasten your seatbelts A small plane takes off from an airport and rises uniformly at an angle of 6° with the horizontal ground. After it has traveled over a horizontal distance of 800m, what is the altitude of the plane to the nearest meter? x 800m 6°

Emergency!!! A ladder on a fire truck can be turned to a maximum angle of 70° and can be extended to a maximum length of 25m. If the base of the ladder is mounted on the fire truck 2m above the ground, how high above the ground will the ladder reach? 2m 25m 70°

Example 4 To measure the width of a river you plant a stake on one side of the river, directly across from a boulder. You then walk 100 meters to the right of the stake and measure a 79° angle between the stake and the boulder. What is the width of the river?

Draw the picture! 5. You lean a ladder 6.7 m long against the wall. It makes an angle of 63º with the level ground. How high up is the top of the ladder? 6. You must order a new rope for the flagpole. To find out what length of rope is needed, you observe that the pole casts a shadow 11.6 m long on the ground. The angle of elevation of the sun is 36º. How tall is the pole? 7. Your cat is trapped on a tree branch 6.5 m above the ground. Your ladder is only 6.7 m long. If you place the ladder’s tip on the branch, what angle will the ladder make with the ground?

8. If the 622 ft Tower of the Americas in San Antonio casts a 260 ft shadow, what is the angle of elevation of the sun? 9. A ground observer, 5 km from the space shuttle launch pad, watches the shuttle climb into the sky. The space shuttle’s instruments report that it is 2 km above the ground. What is the shuttle’s angle of elevation from the observer at that moment? An observer in an airplane at a height of 500 m sees a car at an angle of depression of 31 degrees. If the plane is over a barn, how far is the car from the barn?

11. An observer sees a kite at an angle of elevation of 39 degrees 11. An observer sees a kite at an angle of elevation of 39 degrees. The kite is directly above a tree that is 112 ft from the observer. Find the height of the kite above the ground if the observer is 6 ft tall. 12. An observer at the top of a 50 m lighthouse sees two boats approaching, one behind the other. The angles to the boats are 39 degrees and 25 degrees. Find the distance between the boats.

A couple “special” Applications Squares are made up of two 45-45-90 triangles (when a diagonal is drawn in) An equilateral triangle can be divided into two 30-60-90 triangles to find the height

Example Find the length of a diagonal of a square with side length 5

Example Find the altitude of an equilateral triangle with side length 4

Example Find the perimeter of a square with diagonal length 10