Mathematics 10 — Exercise 7.2 · Q1 (iv)

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.

   $$\sin \left(\mathrm{A°}\right)=\frac{4.0\times \sin \left(\mathrm{71.0°}\right)}{4.8}$$

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

   $$\sin \left(\mathrm{A°}\right)=0.78793215$$

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

   $$(51.992685,128.00731)$$

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

   $$C=57.007315$$

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

   $$128.00731$$

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

   $$c=\frac{4.8\times \sin \left(\mathrm{57.00731489182614°}\right)}{\sin \left(\mathrm{71.0°}\right)}$$

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

   $$c=\frac{4.0259525}{0.94551858}$$

8. Divide by the denominator sine.

   $$c=4.2579306$$

Answer

The final answer is shown below.