Question

which equation is true for the triangle?
a $\tan(55^{circ})=\frac{c}{a}$
b $sin(55^{circ})=\frac{a}{c}$
c $cos(55^{circ})=\frac{c}{b}$
d $cos(55^{circ})=\frac{a}{b}$

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. For the $55^{\circ}$ angle in the given right - triangle, the opposite side is $a$, the adjacent side is $b$, and the hypotenuse is $c$.

Step2: Analyze each option

• Option A: $\tan(55^{\circ})=\frac{a}{b}

eq\frac{c}{a}$.

• Option B: $\sin(55^{\circ})=\frac{a}{c}$. This is correct as per the definition of sine ($\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$).
• Option C: $\cos(55^{\circ})=\frac{b}{c}

eq\frac{c}{b}$.

• Option D: $\cos(55^{\circ})=\frac{b}{c}

eq\frac{a}{b}$.

Answer:

B. $\sin(55^{\circ})=\frac{a}{c}$