Question

in the figure below, $\triangle xyz \sim \triangle abc$.
find $\cos b$, $\sin b$, and $\tan b$.
round your answers to the nearest hundredth.

Explanation:

Step1: Match ∠B to ∠Y

Since $\triangle XYZ \sim \triangle ABC$, corresponding angles are equal, so $\angle B = \angle Y$. We can calculate trigonometric values for $\angle Y$ to get values for $\angle B$.

Step2: Identify sides for ∠Y

In $\triangle XYZ$, right-angled at $X$:

• Opposite to $\angle Y$: $XZ = 10.8$
• Adjacent to $\angle Y$: $XY = 23.1$
• Hypotenuse: $YZ = 25.5$

Step3: Calculate $\cos B$ (=$\cos Y$)

Cosine = adjacent/hypotenuse
$\cos B = \frac{23.1}{25.5} \approx 0.91$

Step4: Calculate $\sin B$ (=$\sin Y$)

Sine = opposite/hypotenuse
$\sin B = \frac{10.8}{25.5} \approx 0.42$

Step5: Calculate $\tan B$ (=$\tan Y$)

Tangent = opposite/adjacent
$\tan B = \frac{10.8}{23.1} \approx 0.47$

Answer:

$\cos B \approx 0.91$, $\sin B \approx 0.42$, $\tan B \approx 0.47$