Question

w8 version (november 2023)
name:

1. determine the value of $\theta$.

5

Explanation:

Step1: Find the length of the base of the left - hand right - angled triangle

Using the trigonometric relation \(\tan\alpha=\frac{\text{opposite}}{\text{adjacent}}\). For the angle of \(53^{\circ}\) in the left - hand right - angled triangle, if the height is \(h = 22\) m and the base is \(x\), then \(\tan53^{\circ}=\frac{22}{x}\). Since \(\tan53^{\circ}\approx\frac{4}{3}\), we have \(x=\frac{22}{\tan53^{\circ}}\approx\frac{22}{\frac{4}{3}} = 16.5\) m.

Step2: Find the length of the base of the right - hand right - angled triangle

For the angle of \(27^{\circ}\) in the right - hand right - angled triangle, if the height is \(h = 22\) m and the base is \(y\), then \(\tan27^{\circ}=\frac{22}{y}\). Since \(\tan27^{\circ}\approx\frac{1}{2}\), we have \(y=\frac{22}{\tan27^{\circ}}\approx44\) m.

Step3: Find the length of the hypotenuse of the large triangle

The length of the base of the large triangle \(b=x + y\approx16.5+44 = 60.5\) m. Using the Pythagorean theorem \(l=\sqrt{22^{2}+60.5^{2}}=\sqrt{484 + 3660.25}=\sqrt{4144.25}\approx64.38\) m.

Step4: Use the sine rule

In the large triangle, using the sine rule \(\frac{\sin\theta}{16.5}=\frac{\sin(53^{\circ}+27^{\circ})}{64.38}\). Since \(\sin(53^{\circ}+27^{\circ})=\sin80^{\circ}\approx0.9848\), we have \(\sin\theta=\frac{16.5\times0.9848}{64.38}\).
\(\sin\theta=\frac{16.2492}{64.38}\approx0.2524\). Then \(\theta=\sin^{- 1}(0.2524)\approx14.6^{\circ}\)

Answer:

\(\theta\approx14.6^{\circ}\)