Question

the three right triangles below are similar. the acute angles ∠f, ∠j, and ∠r are all approximately measured to be 55.8°. 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 55.8° 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 55.8°\\), \\(\cos 55.8°\\), and \\(\tan 55.8°\\). round your answers to the nearest hundredth. \\(\sin 55.8°=\square\\) \\(\cos 55.8°=\square\\) \\(\tan 55.8°=\square\\)

Explanation:

Part (a)

Step 1: Calculate \(\frac{EF}{DF}\)

• \(EF = 4.16\) cm, \(DF = 7.4\) cm.
• \(\frac{EF}{DF}=\frac{4.16}{7.4}\approx0.56\)

Step 2: Calculate \(\frac{IJ}{HJ}\)

• \(IJ = 16.64\) cm, \(HJ = 29.6\) cm.
• \(\frac{IJ}{HJ}=\frac{16.64}{29.6}\approx0.56\)

Step 3: Calculate \(\frac{QR}{PR}\)

• \(QR = 33.28\) cm, \(PR = 59.2\) cm.
• \(\frac{QR}{PR}=\frac{33.28}{59.2}\approx0.56\)

Part (b)

Step 1: Calculate \(\sin(55.8^\circ)\)

• Using calculator, \(\sin(55.8^\circ)\approx0.83\)

Step 2: Calculate \(\cos(55.8^\circ)\)

• Using calculator, \(\cos(55.8^\circ)\approx0.56\)

Step 3: Calculate \(\tan(55.8^\circ)\)

• Using calculator, \(\tan(55.8^\circ)=\frac{\sin(55.8^\circ)}{\cos(55.8^\circ)}\approx\frac{0.83}{0.56}\approx1.48\)

Answer:

(a) \(\frac{EF}{DF}=\boldsymbol{0.56}\), \(\frac{IJ}{HJ}=\boldsymbol{0.56}\), \(\frac{QR}{PR}=\boldsymbol{0.56}\)
(b) \(\sin(55.8^\circ)=\boldsymbol{0.83}\), \(\cos(55.8^\circ)=\boldsymbol{0.56}\), \(\tan(55.8^\circ)=\boldsymbol{1.48}\)