Question

in the right triangle shown below, which of the following statements is true about ∠a? f. cos a = \frac{12}{13} g. sin a = \frac{12}{13} h. tan a = \frac{12}{13} j. cos a = \frac{13}{12} k. sin a = \frac{13}{12}

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, for an angle $\angle A$, $\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan A=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle A$ in right - triangle $ABC$, the opposite side to $\angle A$ is $BC = 12$ cm, the adjacent side to $\angle A$ is $AC = 5$ cm, and the hypotenuse is $AB=13$ cm.

Step2: Calculate $\cos A$

$\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AC}{AB}=\frac{5}{13}$.

Step3: Calculate $\sin A$

$\sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AB}=\frac{12}{13}$.

Step4: Calculate $\tan A$

$\tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC}=\frac{12}{5}$.

Answer:

G. $\sin A=\frac{12}{13}$