Problem 1 (Top Triangle):

Here are the solutions for the problems: Problem 1 (Top Triangle): We have a right-angled triangle with a hypotenuse of 8 and an angle of $30^\circ$. We need to find the opposite side $x$ and the adjacent side $y$. Step 1: Find $x$ using the sine function. $$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} $$ $$ \sin(30^\circ) = \frac{x}{8} $$ $$ x = 8 \times \sin(30^\circ) $$ $$ x = 8 \times \frac{1}{2} $$ $$ x = 4 $$ Step 2: Find $y$ using the cosine function. $$ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} $$ $$ \cos(30^\circ) = \frac{y}{8} $$ $$ y = 8 \times \cos(30^\circ) $$ $$ y = 8 \times \frac{\sqrt{3}}{2} $$ $$ y = 4\sqrt{3} $$ The values are: $\boxed{x = 4, y = 4\sqrt{3}}$ Problem 2 (Bottom Triangle): We have a right-angled triangle with a hypotenuse of 2 and an angle of $45^\circ$. We need to find the adjacent side $x$ and the opposite side $y$. Step 1: Find $x$ using the cosine function. $$ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} $$ $$ \cos(45^\circ) = \frac{x}{2} $$ $$ x = 2 \times \cos(45^\circ) $$ $$ x = 2 \times \frac{\sqrt{2}}{2} $$ $$ x = \sqrt{2} $$ Step 2: Find $y$ using the sine function. $$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} $$ $$ \sin(45^\circ) = \frac{y}{2} $$ $$ y = 2 \times \sin(45^\circ) $$ $$ y = 2 \times \frac{\sqrt{2}}{2} $$ $$ y = \sqrt{2} $$ The values are: $\boxed{x = \sqrt{2}, y = \sqrt{2}}$ That's 2 down. 3 left today — send the next one.