Question

the three right triangles below are similar. the acute angles ∠j, ∠p, and ∠x are all approximately measured to be 27.6°. the side lengths for each triangle are as follows. note that the triangles are not drawn to scale. three right triangles with labels and side lengths: first triangle jkl: jk=17.48 m, kl=9.15 m, jl=19.73 m, angle at j is 27.6°. second triangle pqr: pq=139.84 m, qr=73.2 m, pr=157.84 m, angle at p is 27.6°. third triangle xyz: xy=157.32 m, yz=82.35 m, xz=177.57 m, angle at x is 27.6°. (a) for each triangle, find the ratio of the length of the side opposite 27.6° to the length of the side adjacent to 27.6°. note that \side adjacent\ cannot also refer to the hypotenuse. round your answers to the nearest hundredth. \\(\frac{kl}{jk} = \square\\) \\(\frac{qr}{pq} = \square\\) \\(\frac{yz}{xy} = \square\\) (b) use the aleks calculator to find \\(\sin 27.6°\\), \\(\cos 27.6°\\), and \\(\tan 27.6°\\). round your answers to the nearest hundredth. \\(\sin 27.6° = \square\\) \\(\cos 27.6° = \square\\) \\(\tan 27.6° = \square\\)

Explanation:

Part (a)

Step 1: Calculate \(\frac{KL}{JK}\)

For the first triangle, \(KL = 9.15\) m and \(JK = 17.48\) m. So, \(\frac{KL}{JK}=\frac{9.15}{17.48}\approx0.52\) (rounded to nearest hundredth).

Step 2: Calculate \(\frac{QR}{PQ}\)

For the second triangle, \(QR = 73.2\) m and \(PQ = 139.84\) m. So, \(\frac{QR}{PQ}=\frac{73.2}{139.84}\approx0.52\) (rounded to nearest hundredth).

Step 3: Calculate \(\frac{YZ}{XY}\)

For the third triangle, \(YZ = 82.35\) m and \(XY = 157.32\) m. So, \(\frac{YZ}{XY}=\frac{82.35}{157.32}\approx0.52\) (rounded to nearest hundredth).

Part (b)

Step 1: Find \(\sin27.6^{\circ}\)

Using a calculator, \(\sin27.6^{\circ}\approx0.46\) (rounded to nearest hundredth).

Step 2: Find \(\cos27.6^{\circ}\)

Using a calculator, \(\cos27.6^{\circ}\approx0.89\) (rounded to nearest hundredth).

Step 3: Find \(\tan27.6^{\circ}\)

Using a calculator, \(\tan27.6^{\circ}=\frac{\sin27.6^{\circ}}{\cos27.6^{\circ}}\approx\frac{0.46}{0.89}\approx0.52\) (rounded to nearest hundredth).

Answer:

(a) \(\frac{KL}{JK}=\boldsymbol{0.52}\), \(\frac{QR}{PQ}=\boldsymbol{0.52}\), \(\frac{YZ}{XY}=\boldsymbol{0.52}\)
(b) \(\sin27.6^{\circ}=\boldsymbol{0.46}\), \(\cos27.6^{\circ}=\boldsymbol{0.89}\), \(\tan27.6^{\circ}=\boldsymbol{0.52}\)