Question

If cosA = 3/5 and A ∈ (0,90), what is sin2A?

Answer 1

Answer:

$\sin(2A)=\frac{24}{25}$

Step-by-step explanation:

Double angle identity for sine is

$\sin(2A)=2\sin(A)\cos(A)$.

We know $\cos(A)$. We now need to find $\sin(A)$ to find our answer.

Recall the following $Pythagorean Identity$:

$\sin^2(A)+\cos^2(A)=1$.

Replace $\cos(A)$ with $\frac{3}{5}$:

$\sin^2(A)+(\frac{3}{5})^2=1$

$\sin^2(A)+\frac{9}{25}=1$

Subtract $\frac{9}{25}$ on both sides:

$\sin^2(A)=1-\frac{9}{25}$

$\sin^2(A)=\frac{25}{25}-\frac{9}{25}$

$\sin^2(A)=\frac{25-9}{25}$

$\sin^2(A)=\frac{16}{25}$

Now to finally get the value of $\sin(A)$ take the square of both sides:

$\sin(A)=\pm \sqrt{\frac{16}{25}}$

Since $A$ is in the interval $(0,90)$, then $\sin(A)>0$.

$\sin(A)=\sqrt{\frac{16}{25}}$

$\sin(A)=\frac{\sqrt{16}}{\sqrt{25}}$

$\sin(A)=\frac{4}{5}$

So let's finally find the numerical value of $\sin(2A)$.

$\sin(2A)=2\sin(A)\cos(A)$

$\sin(2A)=2\cdot \frac{4}{5} \cdot \frac{3}{5}$

$\sin(2A)=\frac{2(4)(3)}{5(5)}$

$\sin(2A)=\frac{24}{25}$

Side note:

If $A$ is between 0 and 90 degrees, then we are in the first quadrant.

If we are in the first quadrant, both $x=\cos(A)$ and $y=\sin(A)$ are positive.

Answer 2

Answer:

$\frac{24}{25}$

Step-by-step explanation:

Given

cosA = $\frac{3}{5}$ = $\frac{adjacent}{hypotenuse}$

Then the opposite side using Pythagorean triple is 4, thus

sinA = $\frac{4}{5}$

Using the trigonometric identity

sin2A = 2 sinAcosA, then

sin2A = 2 × $\frac{4}{5}$ × $\frac{3}{5}$

         = $\frac{2(4)(3)}{5(5)}$ = $\frac{24}{25}$