Applications of Trigonometry practice questions

• Let $H$ m be the height of the cliff and $t$ m be the height of the tower.
• Using the angle of elevation to the top of the cliff:
• $\tan 26° = \dfrac{H}{95} \quad \scriptsize\textit{angle of elevation}$
• $H = 95 \times \tan 26°$
• $H \approx 95 \times 0.4877$
• $H = 46.33$ m (to 2 d.p.)
• Using the angle of elevation to the top of the tower:
• $\tan 44° = \dfrac{H + t}{95} \quad \scriptsize\textit{angle of elevation}$

Answer: $45.4$

• Draw north lines at both $P$ and $Q$ (parallel lines).
• $\angle x = 180° - 167°$ (interior $\angle$s, north lines parallel)
• $\angle x = 13°$
• $\angle PQR = 13° + 67°$
• $\angle PQR = 80°$
• $PR^2 = PQ^2 + QR^2 - 2 \times PQ \times QR \times \cos(\angle PQR) \quad \scriptsize\textit{cosine rule}$
• $PR^2 = 30^2 + 30^2 - 2(30)(30)\cos 80°$
• $PR^2 = 900 + 900 - 1800\cos 80°$

Answer: $38.6$

• Step 1: Find the diagonal of the base.
• $d_1^2 = 4^2 + 3^2$
• $= 25$
• $d_1 = 5$ cm $\quad \scriptsize\textit{base diagonal}$
• Step 2: Find the space diagonal.
• $d^2 = d_1^2 + 9^2$
• $= 25 + 81$
• $= 106$

Answer: $60.9$