Question: FINAL EXAM Page 6: Question 6 (4 points) a) Determine the value of $x$. $x$ $18^\circ$ $48m$ Graph shape: a right triangle in the center-top area with the right angle at the upper-left vertex, $x$ marked on the left side, $18^\circ$ marked near the right vertex, and $48m$ labeled along the long bottom side. b) Determine the measure of $\angle A$ to the nearest degree. $A$ $B$ $3m$ $7m$ $C$ Graph shape: a right triangle in the center-lower area with the right angle at vertex $B$, side $AB$ labeled $3m$, side $AC$ labeled $7m$, and point $C$ at the bottom. Show all your work!

1. **Problem Statement:** We have two right triangles. (a) Find the length $x$ in the first triangle with a right angle at the upper-left vertex, an angle of $18^\circ$, and hypotenuse $48m$. (b) Find the measure of $\angle A$ in the second triangle with right angle at $B$, sides $AB=3m$ and $AC=7m$. 2. **Formulas and Rules:** - In a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ - The cosine of an angle is the ratio of the adjacent side to the hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ - The tangent of an angle is the ratio of the opposite side to the adjacent side: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ - To find an angle given two sides, use the inverse trigonometric functions, e.g., $$\theta = \arcsin(\frac{\text{opposite}}{\text{hypotenuse}})$$ or $$\theta = \arctan(\frac{\text{opposite}}{\text{adjacent}})$$. 3. **Part (a) Find $x$:** - Given: - Right angle at upper-left vertex - Angle near right vertex is $18^\circ$ - Hypotenuse (long bottom side) = $48m$ - Side $x$ is opposite the $18^\circ$ angle - Using sine: $$\sin(18^\circ) = \frac{x}{48}$$ - Solve for $x$: $$x = 48 \times \sin(18^\circ)$$ - Calculate $\sin(18^\circ)$: $$\sin(18^\circ) \approx 0.3090$$ - Therefore: $$x = 48 \times 0.3090 = 14.832$$ - Rounded to two decimal places: $$x \approx 14.83m$$ 4. **Part (b) Find $\angle A$:** - Given: - Right angle at $B$ - Side $AB = 3m$ (adjacent to $\angle A$) - Side $AC = 7m$ (hypotenuse) - Use cosine to find $\angle A$: $$\cos(\angle A) = \frac{AB}{AC} = \frac{3}{7}$$ - Calculate $\angle A$: $$\angle A = \arccos\left(\frac{3}{7}\right)$$ - Calculate the value: $$\frac{3}{7} \approx 0.4286$$ - Using inverse cosine: $$\angle A \approx \arccos(0.4286) \approx 64.62^\circ$$ - Rounded to nearest degree: $$\angle A \approx 65^\circ$$ **Final answers:** - (a) $x \approx 14.83m$ - (b) $\angle A \approx 65^\circ$