1. **Problem 1:** Given a right triangle with an angle of 30°, hypotenuse 24 cm, and opposite side labeled 24 cm, solve for $x$ which is the adjacent side. 2. **Choosing the trigonometric function:** Since we know the hypotenuse and want to find the adjacent side, we use cosine, because cosine relates adjacent and hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 3. **Set up the equation:** $$\cos(30^\circ) = \frac{x}{24}$$ 4. **Solve for $x$:** $$x = 24 \times \cos(30^\circ)$$ 5. **Calculate $\cos(30^\circ)$:** $$\cos(30^\circ) = \frac{\sqrt{3}}{2}$$ 6. **Substitute and simplify:** $$x = 24 \times \frac{\sqrt{3}}{2} = \cancel{24} \times \frac{\sqrt{3}}{\cancel{2}} \times 12 = 12\sqrt{3}$$ 7. **Final answer:** $$x = 12\sqrt{3} \approx 20.78 \text{ cm}$$ --- **Problem 2:** Given right triangle ABC with right angle at B, hypotenuse AC = 24 cm, angle A = 30°, and side BC labeled $x$ (adjacent to angle A), solve for angle $C$. Since the triangle's angles sum to 180° and one angle is 90°, the other two angles sum to 90°: $$\angle C = 90^\circ - 30^\circ = 60^\circ$$ --- **Problem 3:** Decide whether to use sine, cosine, tangent, or Pythagorean theorem to find $x$ in a right triangle. If $x$ is a side and you know two sides, use Pythagorean theorem: $$a^2 + b^2 = c^2$$ If you know an angle and one side, use sine, cosine, or tangent depending on which sides are known: - Sine: opposite/hypotenuse - Cosine: adjacent/hypotenuse - Tangent: opposite/adjacent Use the function that relates the known sides and the unknown $x$. --- **Summary:** - For Problem 1, use cosine to find $x$. - For Problem 2, find angle $C$ by subtracting from 90°. - For Problem 3, choose the method based on known sides and angles.