Mathematics 10 — Exercise 7.2 · Q2 (i)

Full question

Calculate the area of triangle ABC using the following measurements.

Step-by-step solution

1. Substitute the given values into the displayed formula.

   $$c=\sqrt{{7.0}^2+{8.0}^2-8.0\times 14.0\times \cos \left(\mathrm{38.0°}\right)}$$

2. Evaluate the powers and trigonometric values before the final arithmetic.

   $$c=4.9742131$$

3. Round only the final result, keeping full precision for later calculations.

   $$c=\mathrm{4.974\; cm}$$

4. Use cosine law to find this angle’s cosine.

   $$\cos \left(\mathrm{A°}\right)=\frac{{8.0}^2+{4.9742131}^2-49.0}{2\times 8.0\times 4.9742131}$$

5. Calculate the squared terms, subtract, then divide by the denominator.

   $$\cos \left(\mathrm{A°}\right)=0.49936034$$

6. Take inverse cosine in DEGREE mode.

   $$A=60.042311$$

7. Use cosine law to find this angle’s cosine.

   $$\cos \left(\mathrm{B°}\right)=\frac{{7.0}^2+{4.9742131}^2-64.0}{2\times 7.0\times 4.9742131}$$

8. Calculate the squared terms, subtract, then divide by the denominator.

   $$\cos \left(\mathrm{B°}\right)=0.13990434$$

9. Take inverse cosine in DEGREE mode.

   $$B=81.957689$$

10. Subtract A and B from 180° to find C.

    $$C=38.0$$

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

    $$\mathrm{area}=\frac{56.0\times \sin \left(\mathrm{37.999999999999986°}\right)}{2}$$

12. Evaluate the sine first.

    $$\sin \left(\mathrm{37.999999999999986°}\right)=0.61566147$$

13. Multiply the two side lengths.

    $$7.0\times 8.0=56.0$$

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

    $$\frac{34.477043}{2}$$

15. The area is in square centimetres.

    $$\mathrm{area\; (cm²)}=17.238521$$

Answer

The final answer is shown below.