**Finding Angles in Right Triangles**

**I**n this lesson, we explore a variety of word problems in which we must calculate the measure of an angle from a right triangle. With inverse trigonometric functions sin^{–1}, cos^{–1}, and tan^{–1}, we can find the angle of a right triangle given any two sides (see Figure 15.1).

**Example 1 **

A 20-foot board is leaning against a wall. If the board reaches 17 feet up the wall, what are the angles that the board makes with the floor and the wall?

The angle opposite the 17-foot wall will have . A calculator shows this to be approximately 58.2°. The other angle must be the complement (make them add to 90°), 90 – 58.2 = 31.8°. This is shown in Figure 15.2.

In the last example, we knew the length of the diagonal hypotenuse, and so we used sine. For most problems, we know the lengths of the two legs, and so we use tangent.

**Example 2**

Suppose you want to aim exactly 20 feet forward and 12 feet to the right. At what angle should you turn? (see Figure 15.3).

**Example 3**

Asecurity camera is installed in a museum hallway. It is 10 feet away from the middle of a 30-foot section of wall that it must monitor (see Figure 15.4). How many degrees should it pan to the left and to the right?

In this case, the side opposite the angle *θ* is *O* = 15 feet, so . Thus, the camera should turn 56.3° in each direction.

**Example 4**

A beam is to be cut to strengthen a wooden rectangle, as illustrated in Figure 15.5. At what angles *θ*_{1} and *θ*_{2} should the two ends of the beam be cut? Suppose that the space inside the rectangle is 12 feet by 9 feet 4 inches.

For angle *θ*_{1} the opposite length is *O* = 9'4" = 9.333 feet and the adjacent side is *A* = 12 feet. Thus, . Angle *θ*_{2} is the complement, *θ*_{2} ≈ 90 – 37.87 = 52.13°. Of course, we could always calculate for the same answer.

Practice problems for this study guide can be found at:

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