Mathematics 10 — Exercise Review · Q3 (i)

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.

   $$\sin \left(\mathrm{B°}\right)=\frac{3.4\times \sin \left(\mathrm{49.0°}\right)}{5.4}$$

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

   $$\sin \left(\mathrm{B°}\right)=0.47518751$$

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

   $$(28.37156,151.62844)$$

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

   $$C=102.62844$$

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

   $$151.62844$$

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

   $$c=\frac{5.4\times \sin \left(\mathrm{102.6284402876383°}\right)}{\sin \left(\mathrm{49.0°}\right)}$$

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

   $$c=\frac{5.2693651}{0.75470958}$$

8. Divide by the denominator sine.

   $$c=6.9819773$$

Answer

The final answer is shown below.