Question

in the triangle below, \\( \frac { 4 } { 5 } \\) represents which ratio?
\\( \sin c \\)
\\( \sin b \\)
\\( \cos c \\)
\\( \tan b \\)

Explanation:

Step1: Recall SOHCAHTOA

SOHCAHTOA: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Analyze angle C

For $\angle C$:

• Opposite side to $\angle C$: $AB = 4$
• Adjacent side to $\angle C$: $AC = 3$
• Hypotenuse: $BC = 5$

Step3: Calculate $\sin C$

$\sin C=\frac{\text{opposite to } C}{\text{hypotenuse}}=\frac{AB}{BC}=\frac{4}{5}$.

Step4: Check other options (briefly)

• $\sin B$: Opposite to $B$ is $AC = 3$, hypotenuse $5$, so $\sin B=\frac{3}{5}

eq\frac{4}{5}$.

• $\cos C$: $\frac{\text{adjacent to } C}{\text{hypotenuse}}=\frac{AC}{BC}=\frac{3}{5}

eq\frac{4}{5}$.

• $\tan B$: $\frac{\text{opposite to } B}{\text{adjacent to } B}=\frac{AC}{AB}=\frac{3}{4}

eq\frac{4}{5}$.

Answer:

$\sin C$ (the option corresponding to $\sin C$)