Question

If $\tan \left(\theta \right)=3$, find the value of $\sin \left(\theta \right)$.

localid="1648455893738" $\left(A\right)\frac{3}{10}\left(B\right)\frac{3\sqrt{10}}{10}\left(C\right)\frac{10}{3}\left(D\right)\frac{1}{3}$

Short answer

The value of $\sin \left(\theta \right)=\frac{3\sqrt{10}}{10}$.Hence,option (B) is correct.

Step-by-step solution

Step 1. Write down the given information.

It is given that $\tan \left(\theta \right)=3$

Step 2. Calculation.

Since $\tan \left(\theta \right)=3$, therefore construct the triangle ABC having angle $C=\theta$, base $BC=1$, perpendicular height $AB=3$and angle$B=90°$.

Now, by Pythagoras Theorem,

$\begin{array}{c}AC=\sqrt{A{B}^2+B{C}^2}\\ =\sqrt{3^2+1^2}\\ =\sqrt{10}\end{array}$

Now in right triangle ABC by trigonometric ratios,

$\begin{array}{c}\sin \left(C\right)=\frac{AB}{AC}\\ \sin \left(\theta \right)=\frac{3}{\sqrt{10}}\\ \sin \left(\theta \right)=\frac{3\sqrt{10}}{10}\end{array}$

Since, the value of $\sin \left(\theta \right)=\frac{3\sqrt{10}}{10}$. Hence, option (B) is correct.

Step 3. Conclusion.

The value of $\sin \left(\theta \right)=\frac{3\sqrt{10}}{10}$. Hence, option (B) is correct.