Question

Equivalent expression of sin(arctan (x/3))

Answer 1

Answer:

$\frac{x}{\sqrt{x^2+9}}$

Step-by-step explanation:

Let $u=\arctan(\frac{x}{3})$.

Then $\tan(u)=\frac{x}{3}$.

Construct a right triangle.

Make one of the non-right angles $u$.

Recall $\tan(u)$ is equal to the ratio of the side that is opposite to $u$ to the side that is adjacent to $u$.

This means opposite side $=x$ while adjacent side$=3$ since $\tan(u)=\frac{x}{3}=\frac{\text{opposite}}{\text{adjacent}}$.

We can find the hypotenuse by using the Pythagorean Theorem.

$(x)^2+(3)^2=(\text{hyp})^2$

$x^2+9=(\text{hyp})^2$

Take the square root of both sides:

$\sqrt{x^2+9}=\text{hyp}$.

Recall $\sin(u)$ is equal to the ratio of the side that is opposite to $u$ to the hypotenuse.

$\sin(\arctan(\frac{x}{3}))$

$\sin(u)$ -->Because we let the inside equal $u$.

$\frac{x}{\sqrt{x^2+9}}$ -->Because $\sin(u)=\frac{\text{opp}}{\text{hyp}}$