1. Problem 19 asks for the best trigonometric tool to solve for angle $\theta$ in a right triangle with sides opposite $\theta = 8$, adjacent to $\theta = 11$, and hypotenuse $= 12$. 2. The primary trigonometric ratios are sine, cosine, and tangent, defined as: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$ 3. Since all three sides are known, and the triangle is right-angled, the Pythagorean theorem could verify side lengths but is not needed to find $\theta$. 4. The Law of Cosines and Law of Sines are generally used for non-right triangles or when angles and sides are mixed. 5. Here, using one of the primary trigonometric ratios is best. For example, using tangent: $$\tan \theta = \frac{8}{11}$$ 6. Then solve for $\theta$: $$\theta = \tan^{-1}\left(\frac{8}{11}\right)$$ 7. Problem 20 asks for the best trigonometric tool to solve for side $e$ in triangle $\triangle EDF$ with angles $\angle E = 80^\circ$, $\angle F = 45^\circ$, and side $EF = 15$ cm. 8. Since two angles and one side are known, the Law of Sines is the best tool: $$\frac{e}{\sin 45^\circ} = \frac{15}{\sin 55^\circ}$$ 9. Note $\angle D = 180^\circ - 80^\circ - 45^\circ = 55^\circ$. 10. Solve for $e$: $$e = \frac{15 \sin 45^\circ}{\sin 55^\circ}$$ Final answers: - For 19: b) One of the Primary Trigonometric Ratios - For 20: c) The Sine Law