Mathematics 10 — Exercise 7.2 · Q1 (iii)

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.

   $$\sin \left(\mathrm{B°}\right)=\frac{5.2\times \sin \left(\mathrm{48.0°}\right)}{5.0}$$

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

   $$\sin \left(\mathrm{B°}\right)=0.77287062$$

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

   $$(50.612372,129.38763)$$

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

   $$A=81.387628$$

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

   $$A=2.6123724$$

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

   $$a=\frac{5.2\times \sin \left(\mathrm{81.38762761171841°}\right)}{\sin \left(\mathrm{50.61237238828158°}\right)}$$

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

   $$a=\frac{5.1413651}{0.77287062}$$

8. Divide by the denominator sine.

   $$a=6.6522973$$

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

   $$a=\frac{5.2\times \sin \left(\mathrm{2.6123723882815852°}\right)}{\sin \left(\mathrm{129.38762761171841°}\right)}$$

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

    $$a=\frac{0.23700926}{0.77287062}$$

11. Divide by the denominator sine.

    $$a=0.30666098$$

Answer

The final answer is shown below.