Question

the three right triangles below are similar. the acute angles ∠f, ∠j, and ∠r are all approximately measured to be 58.8°. the side lengths for each triangle are as follows. note that the triangles are not drawn to scale. (a) for each triangle, find the ratio of the length of the side adjacent to 58.8° to the length of the hypotenuse. note that \side adjacent\ cannot also refer to the hypotenuse. round your answers to the nearest hundredth. \\(\frac{ef}{df} = \square\\) \\(\frac{ij}{hj} = \square\\) \\(\frac{qr}{pr} = \square\\) (b) use the aleks calculator to find \\(\sin 58.8°\\), \\(\cos 58.8°\\), and \\(\tan 58.8°\\). round your answers to the nearest hundredth. \\(\sin 58.8° = \square\\) \\(\cos 58.8° = \square\\) \\(\tan 58.8° = \square\\)

Explanation:

Step1: Calculate \( \frac{EF}{DF} \)

EF = 10.32 m, DF = 19.93 m.
\( \frac{EF}{DF} = \frac{10.32}{19.93} \approx 0.52 \) (rounded to nearest hundredth).

Step2: Calculate \( \frac{IJ}{HJ} \)

IJ = 30.96 m, HJ = 59.79 m.
\( \frac{IJ}{HJ} = \frac{30.96}{59.79} \approx 0.52 \) (rounded to nearest hundredth).

Step3: Calculate \( \frac{QR}{PR} \)

QR = 72.24 m, PR = 139.51 m.
\( \frac{QR}{PR} = \frac{72.24}{139.51} \approx 0.52 \) (rounded to nearest hundredth).

Step4: Find \( \sin 58.8^\circ \), \( \cos 58.8^\circ \), \( \tan 58.8^\circ \)

Using calculator:
\( \sin 58.8^\circ \approx 0.86 \), \( \cos 58.8^\circ \approx 0.52 \), \( \tan 58.8^\circ \approx 1.65 \) (rounded to nearest hundredth).

Answer:

(a) \( \frac{EF}{DF} = \boldsymbol{0.52} \), \( \frac{IJ}{HJ} = \boldsymbol{0.52} \), \( \frac{QR}{PR} = \boldsymbol{0.52} \)
(b) \( \sin 58.8^\circ = \boldsymbol{0.86} \), \( \cos 58.8^\circ = \boldsymbol{0.52} \), \( \tan 58.8^\circ = \boldsymbol{1.65} \)