To solve the triangle ABC with the given data, we will use the Law of Sines and the fact that the sum of the angles in a triangle is $180^\circ$.

Given:

• $\angle A = 52^\circ$
• $\angle C = 67^\circ$
• $a = 14$ (side opposite to angle $A$)

Step 1: Calculate $\angle B$

The sum of the angles in a triangle is $180^\circ$. Thus, $\angle B = 180^\circ - \angle A - \angle C$ $\angle B = 180^\circ - 52^\circ - 67^\circ$ $\angle B = 61^\circ$

Step 2: Use the Law of Sines to find side $b$

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

First, we find $b$: $\frac{14}{\sin 52^\circ} = \frac{b}{\sin 61^\circ}$

Rearrange to solve for $b$: $b = 14 \cdot \frac{\sin 61^\circ}{\sin 52^\circ}$

Using a calculator to find the sines: $\sin 52^\circ \approx 0.7880$ $\sin 61^\circ \approx 0.8746$

Now calculate $b$: $b = 14 \cdot \frac{0.8746}{0.7880} \approx 15.52$

Step 3: Use the Law of Sines to find side $c$

Similarly, for $c$: $\frac{14}{\sin 52^\circ} = \frac{c}{\sin 67^\circ}$

Rearrange to solve for $c$: $c = 14 \cdot \frac{\sin 67^\circ}{\sin 52^\circ}$

Using a calculator to find the sine: $\sin 67^\circ \approx 0.9205$

Now calculate $c$: $c = 14 \cdot \frac{0.9205}{0.7880} \approx 16.35$

Summary of results

• $\angle B = 61^\circ$
• $b \approx 15.52$
• $c \approx 16.35$

Would you like details on the steps or have any other questions?

Here are some related questions:

1. How would the solution change if $\angle C$ was different?
2. Can you solve the triangle using only the Law of Cosines?
3. How do you verify the accuracy of the calculated sides $b$ and $c$?
4. What is the significance of the Law of Sines in solving triangles?
5. How would the solution approach change if only two sides and one angle were given?
6. What other methods can be used to solve triangles in trigonometry?
7. How does the sum of angles property help in solving triangles?
8. What is the relationship between the sides and angles in any given triangle?

Tip: When solving triangles, always double-check your angle measures to ensure their sum is $180^\circ$. This is a quick way to catch errors early.