Mathematics 10 — Exercise 7.2 · Q2 (v)

Full question

Calculate the area of triangle ABC using the following measurements.

Step-by-step solution

1. Angles of a triangle sum to 180°. Subtract both known angles.

   $$A=180-36.0-80.0$$

2. Calculate the remaining angle.

   $$A=64.0$$

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

   $$a=\frac{4.0\times \sin \left(\mathrm{64.0°}\right)}{\sin \left(\mathrm{80.0°}\right)}$$

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

   $$a=\frac{3.5951762}{0.98480775}$$

5. Divide by the denominator sine.

   $$a=3.6506376$$

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

   $$b=\frac{4.0\times \sin \left(\mathrm{36.0°}\right)}{\sin \left(\mathrm{80.0°}\right)}$$

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

   $$b=\frac{2.351141}{0.98480775}$$

8. Divide by the denominator sine.

   $$b=2.3874111$$

9. Use half the product of two sides and the sine of their INCLUDED angle.

   $$\mathrm{area}=\frac{8.7155728\times \sin \left(\mathrm{80.0°}\right)}{2}$$

10. Evaluate the sine first.

    $$\sin \left(\mathrm{80.0°}\right)=0.98480775$$

11. Multiply the two side lengths.

    $$3.6506376\times 2.3874111=8.7155728$$

12. Multiply by the sine, then divide by 2.

    $$\frac{8.5831637}{2}$$

13. The area is in square centimetres.

    $$\mathrm{area\; (cm²)}=4.2915819$$

Answer

The final answer is shown below.