CK-12-Pre-Calculus Concepts

4.8. Applications of Basic Triangle Trigonometry http://www.ck12.org

In the upper right corner of the picture there are four important angles that are marked with angles. The measures
of these angles from the outside in are 90◦, 45 ◦, 21 ◦, 69 ◦. There is a 45-45-90 right triangle on the right, so the base
must also be 50. Therefore you can set up and solve an equation forx.

tan 69◦=x+ 5050

x=50 tan 69◦− 50 ≈ 80. 25 ...f t

The hardest part of this problem is drawing a picture and working with the angles.

3. 4 miles SW and then 2 miles NNW
   Translate SW and NNW into degrees bearing. SW is a bearing of 225◦and NNW is a bearing of 315◦. Draw a
   picture in two steps. Draw the original 4 miles traveled and draw the second 2 miles traveled from the origin. Then
   translate the second leg of the trip so it follows the first leg. This way you end up with a parallelogram, which has
   interior angles that are easier to calculate.

The angle between the two red line segments is 67. 5 ◦which can be seen if the red line is extended past the origin.
The shorter diagonal of the parallelogram is the required unknown information.

x^2 = 42 + 22 − 2 · 4 · 2 ·cos 67. 5 ◦

x≈ 3. 7 miles

Practice

The angle of depression of a boat in the distance from the top of a lighthouse isπ 6. The lighthouse is 150 feet
tall. You want to find the distance from the base of the lighthouse to the boat.

1. Draw a picture of this situation.


2. What methods or techniques will you use?


3. Solve the problem.
   From the third story of a building (60 feet) Jeff observes a car moving towards the building driving on the streets
   below. The angle of depression of the car changes from 34◦to 62◦while he watches. You want to know how far the
   car traveled.