1. Let's start with a basic trigonometry problem: Find $\sin(30^\circ)$. 2. The formula for sine in a right triangle is $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$. 3. For $30^\circ$, from the special triangle rules, $\sin(30^\circ) = \frac{1}{2}$. 4. This means the side opposite the $30^\circ$ angle is half the length of the hypotenuse. 5. Next, find $\cos(45^\circ)$. 6. The cosine formula is $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$. 7. For $45^\circ$, $\cos(45^\circ) = \frac{\sqrt{2}}{2}$. 8. This comes from the isosceles right triangle where the legs are equal and the hypotenuse is $\sqrt{2}$ times a leg. 9. Finally, find $\tan(60^\circ)$. 10. The tangent formula is $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. 11. For $60^\circ$, $\tan(60^\circ) = \sqrt{3}$. 12. This is from the special triangle where the opposite side is $\sqrt{3}$ times the adjacent side. These exercises help understand the values of sine, cosine, and tangent for common angles using special triangles and their ratios.