What does the graph of $\sin^2 x$ look like?

Question

When solving trig equations, sometimes it comes out as a hidden quadratic, like this:

$2\sin^2x-5\sin x+2=0$

Obviously it is possible to factorise and solve for $\sin x$.

I understand graph transformations so $-\sin x$ is an inverted sine graph, so what would $\sin^2x$ look like? As the sine function itself is being raised to the power $2$.

Answer 1

A neat way to think of this is by noticing that

$$ \cos(2a) = 1 - 2\sin^2(a) $$

Hence,

$$ \sin^2(a) = \frac{1 - \cos(2a)}{2} $$

So, it's the graph of $\cos(a)$ flipped, "sped up" by a factor of 2, raised up by 1 unit above the $y$-axis, and then finally shrunk by a factor of $2$ along the $y$-axis.

WolframAlpha plot for reference

Answer 2

1. If the sine value is negative than the square of it is positive.

2. Since sine is periodic in 0 to pi, its square is also periodic but with period pi.

3. Also since sine value lies from $-1$ to $1$ its square lies from $0$ to $1$

4. Square of a number between 0 and 1 will be still lesser than the number.