Mathematics 10 — Exercise 7.2 · Q1 (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.

   $$\sin \left(\mathrm{B°}\right)=\frac{8.4\times \sin \left(\mathrm{42.0°}\right)}{6.1}$$

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

   $$\sin \left(\mathrm{B°}\right)=0.92142575$$

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

   $$(67.135415,112.86458)$$

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

   $$C=70.864585$$

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

   $$C=25.135415$$

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

   $$c=\frac{6.1\times \sin \left(\mathrm{70.86458460526921°}\right)}{\sin \left(\mathrm{42.0°}\right)}$$

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

   $$c=\frac{5.7629535}{0.66913061}$$

8. Divide by the denominator sine.

   $$c=8.6125989$$

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

   $$c=\frac{6.1\times \sin \left(\mathrm{25.135415394730785°}\right)}{\sin \left(\mathrm{42.0°}\right)}$$

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

    $$c=\frac{2.5910304}{0.66913061}$$

11. Divide by the denominator sine.

    $$c=3.8722342$$

Answer

The final answer is shown below.