Mathematics 10 — Exercise 7.2 · Q1 (viii)

Full question

Using the following measurements, solve triangle ABC by finding all its unknown sides and angles. If two triangles are possible, give both solutions.

Step-by-step solution

1. Angles of a triangle sum to 180°. Subtract both known angles.

   $$C=180-92.0-77.0$$

2. Calculate the remaining angle.

   $$C=11.0$$

3. Rearrange the sine law to isolate the missing side.

   $$a=\frac{12.0\times \sin \left(\mathrm{92.0°}\right)}{\sin \left(\mathrm{77.0°}\right)}$$

4. Evaluate both sines and multiply the known side by the numerator sine.

   $$a=\frac{11.99269}{0.97437007}$$

5. Divide by the denominator sine.

   $$a=12.308147$$

6. Rearrange the sine law to isolate the missing side.

   $$c=\frac{12.0\times \sin \left(\mathrm{11.0°}\right)}{\sin \left(\mathrm{77.0°}\right)}$$

7. Evaluate both sines and multiply the known side by the numerator sine.

   $$c=\frac{2.2897079}{0.97437007}$$

8. Divide by the denominator sine.

   $$c=2.3499367$$

Answer

The final answer is shown below.