Unit A: Geometry

Chapter 3: Trigonometry

Finding an Angle in a Triangle Using the Cosine Law

The cosine law can be used to find an angle given the three side lengths. The cosine law can be rearranged to find the angle more easily.

\cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c} \cos B = \frac{a^{2} + c^{2} - b^{2}}{2 a c} \cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}

These three versions are identical. Use the version of the cosine law based on the angle that must be calculated.

Recall: The side and angle opposite each other are given the same letter. Sides are given the lowercase letter, and angles are given the uppercase letter. For example, side

a

is opposite

∠ A

Example 3

Use the cosine law to find

∠ A

Solution

Label ΔABC.

Note: the triangle can be labelled differently from the diagram below as long as matching pairs are used

∠ A

is opposite side a

∠ B

is opposite side b

∠ C

is opposite side c

The given information in ΔABC is

a = 78 ft
b = 91 ft
c = 60 ft

Substitute the known sides into the cosine law to solve for

∠ A

. This version of the cosine law is used because

∠ A

must be found.

\begin{array}{rcl} \cos A & = & \frac{b^{2} + c^{2} - a^{2}}{2 b c} \\ \cos A & = & \frac{91^{2} + 60^{2} - 78^{2}}{2 (91) (60)} \\ = & \frac{8 281 + 3 600 - 6 084}{10 920} \end{array}

Simplify the numerator and denominator before dividing.

\begin{array}{rcl} \cos A & = & \frac{5 797}{10 920} \\ = & 0.5309 \\ ∠ A & = & \cos^{- 1} 0.5309 \\ = & 58 ° \end{array}

Hint: If cos A is not between –1 and 1, an error has occurred.
Recall: If the value is between 0 and 1, the angle is acute. If the value is between –1 and 0, the angle is obtuse.

∠ A

has an angle measure of 58°.

Example 4

Use the cosine law to find

∠ Q

Solution

Label ΔABC.

The given information in ΔABC is

a = 66 cm
b = 92 cm
c = 41 cm

Substitute the known sides into the cosine law to solve for

∠ B

. This version of the cosine law was chosen because

∠ B

must be found.

\begin{array}{rcl} \cos B & = & \frac{a^{2} + c^{2} - b^{2}}{2 a c} \\ \cos B & = & \frac{66^{2} + 41^{2} - 92^{2}}{2 (66) (41)} \\ = & \frac{4 356 + 1 681 - 8 464}{5 412} \\ = & \frac{- 2 427}{5 412} \\ = & - 0.4484 \\ ∠ B & = & \cos^{- 1} (- 0.4484) \\ = & 117 ° \end{array}

∠ B = ∠ Q

and has an angle measure of 117°.