1. **Problem a:** Determine side $a$ in triangle ABC with angles $48^\circ$ at A, $32^\circ$ at B, side $53$ opposite $48^\circ$. 2. Use the Law of Sines: $$\frac{a}{\sin 32^\circ} = \frac{53}{\sin 48^\circ}$$ 3. Calculate $a$: $$a = \frac{53 \sin 32^\circ}{\sin 48^\circ}$$ 4. Substitute values: $$a = \frac{53 \times 0.5299}{0.7431} = \frac{28.0847}{0.7431} = 37.8$$ --- 1. **Problem b:** Triangle with angle $106^\circ$ at C, sides $42$ adjacent to $106^\circ$, $30$ opposite $106^\circ$, find side $a$ between $42$ and $30$. 2. Use Law of Cosines: $$a^2 = 42^2 + 30^2 - 2 \times 42 \times 30 \times \cos 106^\circ$$ 3. Calculate: $$a^2 = 1764 + 900 - 2520 \times (-0.2756) = 2664 + 694.9 = 3358.9$$ 4. Find $a$: $$a = \sqrt{3358.9} = 57.96$$ --- 1. **Problem c:** Right triangle with angles $25^\circ$, $55^\circ$, side $12.4$ opposite $25^\circ$, find side $a$ opposite $55^\circ$. 2. Use Law of Sines: $$\frac{a}{\sin 55^\circ} = \frac{12.4}{\sin 25^\circ}$$ 3. Calculate $a$: $$a = \frac{12.4 \times \sin 55^\circ}{\sin 25^\circ} = \frac{12.4 \times 0.8192}{0.4226} = \frac{10.16}{0.4226} = 24.04$$ --- 1. **Problem 46:** Find angle $Q$ in triangle AQR with angle $68^\circ$ at A, side $10$ opposite $Q$, side $5.5$ adjacent to $68^\circ$. 2. Use Law of Sines: $$\frac{10}{\sin Q} = \frac{5.5}{\sin 68^\circ}$$ 3. Solve for $\sin Q$: $$\sin Q = \frac{10 \sin 68^\circ}{5.5} = \frac{10 \times 0.9272}{5.5} = 1.685$$ 4. Since $\sin Q > 1$, check triangle sides or angles; likely a mistake or no triangle possible. --- 1. **Problem 47a:** Right triangle with $A=90^\circ$, $b=9$, $c=7.5$, find $a$ and other angles. 2. Use Pythagoras: $$a = \sqrt{c^2 - b^2} = \sqrt{7.5^2 - 9^2} = \sqrt{56.25 - 81}$$ 3. Negative under root means error; likely $c$ is hypotenuse, so $c=9$, $b=7.5$. 4. Correct: $$a = \sqrt{9^2 - 7.5^2} = \sqrt{81 - 56.25} = \sqrt{24.75} = 4.97$$ 5. Find angles using sine: $$\sin B = \frac{b}{c} = \frac{7.5}{9} = 0.8333 \Rightarrow B = 56^\circ$$ 6. Angle $C = 90^\circ - 56^\circ = 34^\circ$. --- 1. **Problem 48:** Rafters inclined at $25^\circ$ and $63^\circ$, house width $16.7$ m, find shorter rafter length. 2. Use Law of Sines in triangle formed by rafters and width: $$\frac{L_1}{\sin 63^\circ} = \frac{16.7}{\sin (180^\circ - 25^\circ - 63^\circ)} = \frac{16.7}{\sin 92^\circ}$$ 3. Calculate $L_1$: $$L_1 = \frac{16.7 \sin 63^\circ}{\sin 92^\circ} = \frac{16.7 \times 0.8910}{0.9986} = 14.9$$ --- 1. **Problem 49:** Stan is 13 m and 12 m from goal posts 2 m apart, find shooting angle. 2. Use Law of Cosines for angle between lines: $$\theta = \cos^{-1} \left( \frac{13^2 + 12^2 - 2^2}{2 \times 13 \times 12} \right) = \cos^{-1} \left( \frac{169 + 144 - 4}{312} \right) = \cos^{-1} (0.9936) = 6.6^\circ$$ --- 1. **Problem 50:** Two hikers 5 km N50°E and 4 km S30°E, find distance apart. 2. Convert to components and use Law of Cosines: Angle between paths = $50^\circ + 30^\circ = 80^\circ$ $$d = \sqrt{5^2 + 4^2 - 2 \times 5 \times 4 \times \cos 80^\circ} = \sqrt{25 + 16 - 40 \times 0.1736} = \sqrt{41 - 6.94} = \sqrt{34.06} = 5.84$$ --- 1. **Problem 51:** Tree height with angles $27^\circ$ and $38^\circ$ at points 45 m apart. 2. Let height be $h$, distances $x$ and $x-45$: $$\tan 27^\circ = \frac{h}{x}, \quad \tan 38^\circ = \frac{h}{x-45}$$ 3. Equate: $$x \tan 27^\circ = (x-45) \tan 38^\circ$$ 4. Solve for $x$: $$x (0.5095 - 0.7813) = -45 \times 0.7813 \Rightarrow -0.2718 x = -35.16 \Rightarrow x = 129.4$$ 5. Calculate $h$: $$h = 129.4 \times 0.5095 = 65.9$$ --- 1. **Problem 52:** Tower height with angles $28^\circ$ and $32^\circ$, distance between points 9.4 km. 2. Let height $h$, distances $x$ and $9.4 - x$: $$\tan 28^\circ = \frac{h}{x}, \quad \tan 32^\circ = \frac{h}{9.4 - x}$$ 3. Equate: $$x \tan 28^\circ = (9.4 - x) \tan 32^\circ$$ 4. Solve for $x$: $$x (0.5317 + 0.6249) = 9.4 \times 0.6249 \Rightarrow 1.1566 x = 5.87 \Rightarrow x = 5.07$$ 5. Calculate $h$: $$h = 5.07 \times 0.5317 = 2.7$$ --- **Final answers:** - a) $a = 37.8$ - b) $a = 57.96$ - c) $a = 24.04$ - 46) No valid triangle (check data) - 47a) $a=4.97$, $B=56^\circ$, $C=34^\circ$ - 48) Shorter rafter $=14.9$ - 49) Shooting angle $=6.6^\circ$ - 50) Distance between hikers $=5.84$ - 51) Tree height $=65.9$ - 52) Tower height $=2.7$