Question

1. in △xyz, ∠z is a right angle and tan x = $\frac{8}{15}$. what is sin y?

f $\frac{8}{17}$ g $\frac{15}{17}$ h $\frac{17}{15}$ i $\frac{15}{8}$

1. the sides of a rectangle are 25 cm and 8 cm. what is the measure of the angle formed by the short side and a diagonal of the rectangle?

f 17.7° g 18.7° h 71.3° i 72.3°

Explanation:

Step1: Recall tangent - side relationship

In right - triangle \(XYZ\) with \(\angle Z = 90^{\circ}\), \(\tan X=\frac{YZ}{XZ}=\frac{8}{15}\). Let \(YZ = 8k\) and \(XZ = 15k\) (\(k>0\)).

Step2: Find the hypotenuse using Pythagorean theorem

By the Pythagorean theorem \(XY^{2}=YZ^{2}+XZ^{2}\). Substituting \(YZ = 8k\) and \(XZ = 15k\), we get \(XY=\sqrt{(8k)^{2}+(15k)^{2}}=\sqrt{64k^{2}+225k^{2}}=\sqrt{289k^{2}} = 17k\).

Step3: Recall sine - side relationship and find \(\sin Y\)

\(\sin Y=\frac{XZ}{XY}\). Since \(XZ = 15k\) and \(XY = 17k\), then \(\sin Y=\frac{15}{17}\).

for 55:

Step1: Recall tangent in a rectangle

In a rectangle with sides \(a = 8\mathrm{cm}\) and \(b = 25\mathrm{cm}\), if the short side is \(a = 8\mathrm{cm}\), and we consider the right - triangle formed by the short side, the long side, and the diagonal. Let the angle between the short side and the diagonal be \(\theta\). Then \(\tan\theta=\frac{25}{8}\).

Step2: Find the angle using inverse - tangent

\(\theta=\arctan(\frac{25}{8})\approx72.3^{\circ}\)

Answer:

G. \(\frac{15}{17}\)