Question: Question 15 (a) For each triangle, find the ratio of the length of the side opposite 61.2° to the length of the hypotenuse. Round your answers to the nearest hundredth. JK / JL = PQ / PR = XY / XZ = (b) Use the ALEKS Calculator to find sin 61.2°, cos 61.2°, and tan 61.2°. Round your answers to the nearest hundredth. sin 61.2° = cos 61.2° = tan 61.2° = (c) Which trigonometric function gives each ratio of sides in part (a)? sine cosine tangent none of these

1. **State the problem:** We have three right triangles each with an angle of $61.2^\circ$. We need to find the ratio of the side opposite this angle to the hypotenuse for each triangle, then find the sine, cosine, and tangent of $61.2^\circ$, and finally identify which trigonometric function corresponds to the ratios found. 2. **Formula and rules:** The sine of an angle in a right triangle is defined as: $$\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$$ This means the ratio of the side opposite the angle to the hypotenuse is the sine of that angle. 3. **Calculate the ratios for each triangle:** - For triangle $JKL$: $$\frac{JK}{JL} = \frac{9.45}{19.59} \approx 0.4823$$ Rounded to the nearest hundredth: $$0.48$$ - For triangle $PQR$: $$\frac{PQ}{PR} = \frac{75.6}{156.72} \approx 0.4825$$ Rounded to the nearest hundredth: $$0.48$$ - For triangle $XYZ$: $$\frac{XY}{XZ} = \frac{85.05}{176.31} \approx 0.4824$$ Rounded to the nearest hundredth: $$0.48$$ 4. **Use calculator to find trigonometric values:** - $$\sin 61.2^\circ \approx 0.48$$ - $$\cos 61.2^\circ \approx 0.48$$ (calculator value is about 0.4848 but we will confirm below) - $$\tan 61.2^\circ \approx 1.80$$ (calculator value about 1.80) Using a calculator: $$\sin 61.2^\circ \approx 0.88$$ $$\cos 61.2^\circ \approx 0.48$$ $$\tan 61.2^\circ \approx 1.83$$ 5. **Compare ratios to trigonometric functions:** The ratios from part (a) are approximately $0.48$, which matches the cosine value, not sine. 6. **Conclusion:** - The ratio of the side opposite to the hypotenuse is approximately $0.48$ for all triangles. - The sine of $61.2^\circ$ is approximately $0.88$. - The cosine of $61.2^\circ$ is approximately $0.48$. - The tangent of $61.2^\circ$ is approximately $1.83$. - Since the ratio matches the cosine value, the ratio in part (a) corresponds to the cosine function, not sine. **Final answers:** (a) $$\frac{JK}{JL} = 0.48$$ $$\frac{PQ}{PR} = 0.48$$ $$\frac{XY}{XZ} = 0.48$$ (b) $$\sin 61.2^\circ = 0.88$$ $$\cos 61.2^\circ = 0.48$$ $$\tan 61.2^\circ = 1.83$$ (c) The trigonometric function giving the ratio in part (a) is **cosine**.