Question

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

Answer 1

`\sin(2A)=(24)/(25)`

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
`(3)/(5)`:

`\sin^2(A)+((3)/(5))^2=1`

`\sin^2(A)+(9)/(25)=1`

Subtract
`(9)/(25)` on both sides:

`\sin^2(A)=1-(9)/(25)`

`\sin^2(A)=(25)/(25)-(9)/(25)`

`\sin^2(A)=(25-9)/(25)`

`\sin^2(A)=(16)/(25)`

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

`\sin(A)=\pm \sqrt{(16)/(25)}`

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

`\sin(A)=\sqrt{(16)/(25)}`

`\sin(A)=(√(16))/(√(25))`

`\sin(A)=(4)/(5)`

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

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

`\sin(2A)=2\cdot (4)/(5) \cdot (3)/(5)`

`\sin(2A)=(2(4)(3))/(5(5))`

`\sin(2A)=(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

`(24)/(25)`

Explanation:

Given

cosA =
`(3)/(5)` =
`(adjacent)/(hypotenuse)`

Then the opposite side using Pythagorean triple is 4, thus

sinA =
`(4)/(5)`

Using the trigonometric identity

sin2A = 2 sinAcosA, then

sin2A = 2 ×
`(4)/(5)` ×
`(3)/(5)`

=
`(2(4)(3))/(5(5))` =
`(24)/(25)`