Solving Right Triangles

• Given certain measures in a right triangle, we often want to find the other angle and side measures. This is called solving the triangle.

• All of the following examples will use degree measure, so set the mode on your calculator to degree.

c

7 in

a

62

Example 1:Solve the given triangle:

First, solve for θ.

Since the sum of the two acute angles in a right triangle is 90°, we have …

So θ is 28°.

62 90 28

c

7 in

a

62

Now solve for c, which is the hypotenuse.

The side measured 7 in. is the side adjacent to the given 62° angle.

The trigonometric ratio using adjacent and hypotenuse is cosine.

cos 62adj

hyp

c

7 in

a

62

cos 62adj

hyp 7

c

cos 62

7c 14.91

The value of c is approximately 14.91 inches.

c

7 in

a

62

Last, solve for a, which is the side opposite the 62° angle.

The side measured 7 in. is the side adjacent to the given 62° angle.

The trigonometric ratio using opposite and adjacent is tangent.

tan 62opp

adj

c

7 in

a

62

tan 62opp

adj

7

a

7 tan 62a 13.17

The value of a is approximately 13.17 inches.

84.7 cm

62.3 cm

b

Example 2:Solve the given triangle:

First, solve for θ.

Since the hypotenuse and the side adjacent to θ are given, use cosine.

84.7 cm

62.3 cm

b

So θ is approximately 42.6°.

adjcos

hyp 62.3

84.7 0.736

1cos 0.736 42.6

Now solve for α.

84.7 cm

62.3 cm

b

Since the sum of the two acute angles in a right triangle is 90°, we have …

So α is approximately 47.4°

90

42.6 90

47.4

Last, solve for b, which is the side opposite θ.

The side measured 62.3 in. is the side adjacent to θ

The trigonometric ratio using opposite and adjacent is tangent.

tan 42.6opp

adj

84.7 cm

62.3 cm

b

62.3

b

62.3 tan 42.6b 57.3

The value of b is approximately 57.3 cm.

tan 42.6opp

adj

84.7 cm

62.3 cm

b

Note that we could have used the Pythagorean Theorem to solve for side b.