Here are the solutions to the questions: 37. In a triangle, if $a = 10 \text{ cm}$, $A = 30^\circ$, and $B = 60^\circ$, find $b$. Step 1: Use the Sine Rule. The Sine Rule states that for a triangle with sides $a, b, c$ and opposite angles $A, B, C$: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$ Step 2: Substitute the given values into the Sine Rule. $$ \frac{10 \text{ cm}}{\sin 30^\circ} = \frac{b}{\sin 60^\circ} $$ Step 3: Solve for $b$. We know that $\sin 30^\circ = 0.5$ and $\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$. $$ \frac{10}{0.5} = \frac{b}{0.866} $$ $$ 20 = \frac{b}{0.866} $$ $$ b = 20 \times 0.866 $$ $$ b = 17.32 \text{ cm} $$ Step 4: Compare with the given options. The calculated value $17.32 \text{ cm}$ is closest to option B. The final answer is $\boxed{\text{B. } 17.3 \text{ cm}}$. 38. Which formula is the cosine rule? The Cosine Rule relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides $a, b, c$ and angle $A$ opposite side $a$, the formula is: $$ a^2 = b^2 + c^2 - 2bc \cos A $$ Comparing this with the given options: A. $a^2 = b^2 + c^2$ is the Pythagorean theorem. B. $a^2 = b^2 + c^2 - 2bc \cos A$ is the Cosine Rule. C. $a/\sin A = b/\sin B$ is the Sine Rule. D. $\tan A = \text{opposite}/\text{adjacent}$ is a trigonometric ratio for right-angled triangles. The final answer is $\boxed{\text{B. } a^2 = b^2 + c^2 - 2bc \cos A}$. 39. A ladder $5 \text{ m}$ long leans against a wall, forming a $60^\circ$ angle with the ground. What is the height the ladder reaches on the wall? Step 1: Identify the components of the right-angled triangle formed. The ladder is the hypotenuse ($H = 5 \text{ m}$). The angle with the ground is $\theta = 60^\circ$. The height the ladder reaches on the wall is the side opposite to the angle ($\text{Opposite} = h$). Step 2: Use the sine function. The sine function relates the opposite side, hypotenuse, and angle: $$ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} $$ Step 3: Substitute the values and solve for $h$. $$ \sin 60^\circ = \frac{h}{5 \text{ m}} $$ $$ h = 5 \text{ m} \times \sin 60^\circ $$ We know that $\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$. $$ h = 5 \text{ m} \times 0.866 $$ $$ h = 4.33 \text{ m} $$ Step 4: Compare with the given options. The calculated value $4.33 \text{ m}$ is closest to option C. The final answer is $\boxed{\text{C. } 4.3 \text{ m}}$. 40. A man sees the top of a building, an angle of elevation of $30^\circ$. He is standing $20 \text{ m}$ from the foot of the building. What is the height of the building? Step 1: Identify the components of the right-angled triangle formed. The distance from the building is the adjacent side ($\text{Adjacent} = 20 \text{ m}$). The angle of elevation is $\theta = 30^\circ$. The height of the building is the side opposite to the angle ($\text{Opposite} = h$). Step 2: Use the tangent function. The tangent function relates the opposite side, adjacent side, and angle: $$ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} $$ Step 3: Substitute the values and solve for $h$. $$ \tan 30^\circ = \frac{h}{20 \text{ m}} $$ $$ h = 20 \text{ m} \times \tan 30^\circ $$ We know that $\tan 30^\circ = \frac{1}{\sqrt{3}} \approx 0.577$. $$ h = 20 \text{ m} \times 0.577 $$ $$ h = 11.54 \text{ m} $$ Step 4: Compare with the given options. The calculated value $11.54 \text{ m}$ is closest to option C. The final answer is $\boxed{\text{C. } 11.5 \text{ m}}$. Send me the next one 📸