1. The problem involves finding missing sides or angles in right triangles using trigonometric ratios. 2. Recall the basic trigonometric ratios for a right triangle with angle $\alpha$: - $\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \alpha = \frac{\text{opposite}}{\text{adjacent}}$ 3. For the triangle with angle 30° and hypotenuse 100, find the opposite side $y$: - Use $\sin 30^\circ = \frac{y}{100}$ - Since $\sin 30^\circ = \frac{1}{2}$, we have $\frac{1}{2} = \frac{y}{100}$ - Multiply both sides by 100: $100 \times \frac{1}{2} = y$ - So, $y = 50$ 4. For the triangle with angle 60°, hypotenuse 10, and opposite side $y$, find $y$: - Use $\sin 60^\circ = \frac{y}{10}$ - Since $\sin 60^\circ = \frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2} = \frac{y}{10}$ - Multiply both sides by 10: $10 \times \frac{\sqrt{3}}{2} = y$ - So, $y = 5\sqrt{3}$ 5. For the same triangle, find adjacent side $x$: - Use $\cos 60^\circ = \frac{x}{10}$ - Since $\cos 60^\circ = \frac{1}{2}$, we have $\frac{1}{2} = \frac{x}{10}$ - Multiply both sides by 10: $10 \times \frac{1}{2} = x$ - So, $x = 5$ 6. For the triangle with angle 60°, adjacent side 25, and opposite side $x$, find $x$: - Use $\tan 60^\circ = \frac{x}{25}$ - Since $\tan 60^\circ = \sqrt{3}$, we have $\sqrt{3} = \frac{x}{25}$ - Multiply both sides by 25: $25 \sqrt{3} = x$ 7. For the triangle with angle 30°, opposite side 30, and adjacent side $x$, find $x$: - Use $\tan 30^\circ = \frac{30}{x}$ - Since $\tan 30^\circ = \frac{1}{\sqrt{3}}$, we have $\frac{1}{\sqrt{3}} = \frac{30}{x}$ - Cross multiply: $x = 30 \sqrt{3}$ 8. For the triangle with angle 40°, adjacent side 10, and opposite side $z$, find $z$: - Use $\tan 40^\circ = \frac{z}{10}$ - So, $z = 10 \tan 40^\circ$ - Approximate $\tan 40^\circ \approx 0.8391$, so $z \approx 8.391$ 9. For the triangle with angle 75°, adjacent side 75, and opposite side $z$, find $z$: - Use $\tan 75^\circ = \frac{z}{75}$ - So, $z = 75 \tan 75^\circ$ - Approximate $\tan 75^\circ \approx 3.732$, so $z \approx 279.9$ Final answers: - $y = 50$ - $y = 5\sqrt{3}$ - $x = 5$ - $x = 25\sqrt{3}$ - $x = 30\sqrt{3}$ - $z \approx 8.391$ - $z \approx 279.9$