Mathematics 10 — Exercise Review · Q2 (ii)

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.

   $$a=\sqrt{{2.1}^2+{5.0}^2-4.2\times 5.0\times \cos \left(\mathrm{45.0°}\right)}$$

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

   $$a=3.8158561$$

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

   $$a=\mathrm{3.816\; cm}$$

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

   $$\cos \left(\mathrm{A°}\right)=\frac{{2.1}^2+{5.0}^2-14.5607575950825}{2\times 2.1\times 5.0}$$

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

   $$\cos \left(\mathrm{A°}\right)=0.70710678$$

6. Take inverse cosine in DEGREE mode.

   $$A=45.0$$

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

   $$\cos \left(\mathrm{B°}\right)=\frac{{5.0}^2+{3.8158561}^2-4.41}{2\times 5.0\times 3.8158561}$$

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

   $$\cos \left(\mathrm{B°}\right)=0.92117619$$

9. Take inverse cosine in DEGREE mode.

   $$B=22.901357$$

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

    $$C=112.09864$$

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

    $$\mathrm{area}=\frac{8.0132978\times \sin \left(\mathrm{112.09864301584932°}\right)}{2}$$

12. Evaluate the sine first.

    $$\sin \left(\mathrm{112.09864301584932°}\right)=0.92653754$$

13. Multiply the two side lengths.

    $$3.8158561\times 2.1=8.0132978$$

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

    $$\frac{7.4246212}{2}$$

15. The area is in square centimetres.

    $$\mathrm{area\; (cm²)}=3.7123106$$

Answer

The final answer is shown below.