Mathematics 10 — Exercise 7.2 · Q1 (ii)

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. Use the sine law to find the sine of the angle opposite the other given side.

   $$\sin \left(\mathrm{A°}\right)=\frac{12.2\times \sin \left(\mathrm{50.0°}\right)}{15.8}$$

2. Evaluate the sine, multiply the numerator, then divide by the known opposite side.

   $$\sin \left(\mathrm{A°}\right)=0.59150267$$

3. Inverse sine gives one angle; its supplement has the same sine. Check BOTH against the 180° angle sum.

   $$(36.263715,143.73628)$$

4. For this valid branch subtract both known angles from 180°.

   $$B=93.736285$$

5. Reject this supplement: the known angles would sum to at least 180°.

   $$143.73628$$

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

   $$b=\frac{12.2\times \sin \left(\mathrm{93.73628502378355°}\right)}{\sin \left(\mathrm{36.263714976216455°}\right)}$$

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

   $$b=\frac{12.17407}{0.59150267}$$

8. Divide by the denominator sine.

   $$b=20.581597$$

Answer

The final answer is shown below.