Question

in the triangle below, \frac{4}{5} represents which ratio? a b c sinc sinb cosc tanb

Explanation:

Step1: Recall trigonometric ratios

• $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$

Step2: Analyze $\sin C$

• For $\angle C$, the opposite side is $AB = 4$, hypotenuse is $BC=5$.
• So, $\sin C=\frac{AB}{BC}=\frac{4}{5}$

Step3: Analyze $\sin B$

• For $\angle B$, the opposite side is $AC = 3$, hypotenuse is $BC = 5$. So, $\sin B=\frac{AC}{BC}=\frac{3}{5}$

Step4: Analyze $\cos C$

• For $\angle C$, the adjacent side is $AC = 3$, hypotenuse is $BC = 5$. So, $\cos C=\frac{AC}{BC}=\frac{3}{5}$

Step5: Analyze $\tan B$

• For $\angle B$, the opposite side is $AC = 3$, adjacent side is $AB = 4$. So, $\tan B=\frac{AC}{AB}=\frac{3}{4}$

Answer:

$\sin C$ (i.e., the first option)