Question

find cos y, tan y, and sin y. round your answers to the nearest hundredth.

Explanation:

Step1: Identify Similar Triangles

Triangles \(ABC\) and \(XYZ\) are right - angled triangles. Since \(\angle C=\angle X = 90^{\circ}\), and we can assume that \(\triangle ABC\sim\triangle XYZ\) (by AA similarity, as the angles of a right - angled triangle are related to the ratios of their sides). So the ratios of corresponding sides are equal. In \(\triangle ABC\), \(AC = 30.8\), \(BC=14.4\), \(AB = 34\). In \(\triangle XYZ\), \(\angle X = 90^{\circ}\), so for \(\angle Y\), the adjacent side, opposite side and hypotenuse can be determined by the corresponding sides of \(\triangle ABC\). The side adjacent to \(\angle A\) in \(\triangle ABC\) is \(AC\), opposite is \(BC\), and hypotenuse is \(AB\). For \(\angle Y\) in \(\triangle XYZ\), the adjacent side (to \(\angle Y\)) will correspond to \(BC\) (length \(14.4\)), the opposite side will correspond to \(AC\) (length \(30.8\)), and the hypotenuse will correspond to \(AB\) (length \(34\)).

Step2: Calculate \(\cos Y\)

The formula for cosine of an angle in a right - triangle is \(\cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}\). For \(\angle Y\), the adjacent side length is \(14.4\) and the hypotenuse length is \(34\). So \(\cos Y=\frac{14.4}{34}\approx0.42\) (rounded to the nearest hundredth).

Step3: Calculate \(\tan Y\)

The formula for tangent of an angle in a right - triangle is \(\tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}\). For \(\angle Y\), the opposite side length is \(30.8\) and the adjacent side length is \(14.4\). So \(\tan Y=\frac{30.8}{14.4}\approx2.14\) (rounded to the nearest hundredth).

Step4: Calculate \(\sin Y\)

The formula for sine of an angle in a right - triangle is \(\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}\). For \(\angle Y\), the opposite side length is \(30.8\) and the hypotenuse length is \(34\). So \(\sin Y=\frac{30.8}{34}\approx0.91\) (rounded to the nearest hundredth).

Answer:

\(\cos Y\approx0.42\), \(\tan Y\approx2.14\), \(\sin Y\approx0.91\)