Solving a Triangle Using Law of Sines and Law of Cosines

Q9. Solve the triangle given the diagram and the information: angle B = 25°, angle A = 110°, side b = 3.

Background

Topic: Solving Triangles (Law of Sines and Law of Cosines)

This question tests your ability to solve a triangle when given two angles and one side (ASA or AAS case). You will need to find the remaining angle and the lengths of the other two sides.

Key Terms and Formulas

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Sum of angles in a triangle: $A + B + C = 180^\circ$

• Side labels: $a$ is opposite angle $A$, $b$ is opposite angle $B$, $c$ is opposite angle $C$

Step-by-Step Guidance

1. $\text{First, find the missing angle } C:\ \ C = 180^\circ - A - B$

2. $\text{Next, use the Law of Sines to set up ratios for the unknown sides:}$

3. $\frac{a}{\sin A} = \frac{b}{\sin B}$\ \text{and}\ \frac{c}{\sin C} = \frac{b}{\sin B}$

4. $\text{Plug in the known values for angles and side } b = 3$\ \text{to solve for } a \text{ and } c.$

5. $\text{Calculate } \sin A, \sin B, \text{ and } \sin C \text{ to use in your ratios.}$

Try solving on your own before revealing the answer!

Final Answer:

$C = 45^\circ$, $a \approx 6.67$, $c \approx 5.02$

We used the Law of Sines and the sum of angles in a triangle to find all missing values.