20.2 Solving Sides of Triangles Using Trigonometric Functions

Most problems in trigonometry involve finding the length of an unknown side of a triangle when given one angle and one side or finding an unknown angle in a triangle when the lengths of two sides are given. The solution procedure is always the same:

• Sketch the triangle.

• Identify the known and unknown quantities and show on the sketch.

• Decide which trigonometric function to use.

• Set up an equation using a trigonometric function.

• Solve the equation for the unknown quantity.

Procedure to solve trian gle using trigonometry.

The next examples illustrate how to solve for sides of a triangle using trigonometric functions.

Example 20.4A: Solving a Right Triangle

For the triangle shown in Figure 20.17, solve for ? B and the lengths of sides a and c.

Figure 20.17

Solution:

• STEP 1: Find ? B

  Since the sum of the angle measures of a triangle must add to 180, the measure of ? B is:

  
  

  So, ? B = 62 .

• STEP 2: Find side a

  The length of side a can be solved using either ? A or ? B as a reference. If we choose ? A, then the adjacent side is the known side and the opposite side is the unknown side. This means that we can use the tangent function by writing:

  
  

  Rearranging ? a = 12tan 28 .

  Using a scientific calculator,...