Question

trigonometric ratios
which trigonometric ratios are correct for triangle abc? choose three correct answers.
a
b
30°
9
18
60°
c
□ tan(c)=√3
□ tan(b)=2√3/3
□ sin(c)=√3/2

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$ and $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle C = 60^{\circ}$ and $\angle B=30^{\circ}$ in right - triangle $ABC$ with right - angle at $A$.

Step2: Calculate $\tan(C)$

For $\angle C$, the opposite side to $\angle C$ is $AB$ and the adjacent side is $AC$. If $\angle C = 60^{\circ}$, $\tan(C)=\tan(60^{\circ})=\sqrt{3}$.

Step3: Calculate $\tan(B)$

For $\angle B$, the opposite side to $\angle B$ is $AC$ and the adjacent side is $AB$. If $\angle B = 30^{\circ}$, $\tan(B)=\tan(30^{\circ})=\frac{\sqrt{3}}{3}$, not $\frac{2\sqrt{3}}{3}$.

Step4: Calculate $\sin(C)$

For $\angle C$, the opposite side is $AB$ and the hypotenuse is $BC = 18$. $\sin(C)=\sin(60^{\circ})=\frac{\sqrt{3}}{2}$.

Answer:

$\tan(C)=\sqrt{3}$, $\sin(C)=\frac{\sqrt{3}}{2}$