Question

Solve the triangle: for which side a is 20, side c is 15, and angle C is 40°.

Short answer

The unknown values of the triangle are$A\approx 59°,B=81°,b=23.05$

Step-by-step solution

Step 1. Given Information

The given values are$a=20,c=15,C=40°$.

Step 2. Use the Rule of Sines

• By rule of sines, $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$.
• Substitute the given values to find the value of angle A.

$\begin{array}{rcl}\frac{\sin A}{20}& =& \frac{\sin 40}{15}\\ \sin A& =& 20\times \frac{0.6428}{15}\\ & \approx & 0.857\\ A& \approx & 58.98°\end{array}$

Step 3. Use the Angle Sum Property

• Use the sum of angles of a triangle and find the value of andle B.

$\begin{array}{rcl}\angle B& =& 180°-40°-58.98°\\ & \approx & 81°\end{array}$

Step 4. Use the rule of Sines

• Use the rule of sines again to find the length of side b.

$\begin{array}{rcl}\frac{\sin 81}{b}& =& \frac{\sin 40}{15}\\ b& =& \frac{15\sin 81}{\sin 40}\\ & \approx & 23.05\end{array}$