Can the value of sin theta be greater than 1? [closed]

$\begingroup$

For example, is it possible for sin theta to be 1.06 or 1.2 under any circumstances? Or is it possible for it to be exactly 1?

$\endgroup$ 5

2 Answers

$\begingroup$

No and yes.

No:If you let $\theta$ be an angle in a right angled triangle, we know that $\sin(\theta)$ is equal to $\frac{\text{Opposite}}{\text{Hypotenuse}}$. We know that the Hypotenuse is never shorter than the line Opposite the angle $\theta$, so this fraction can never exceed $1$.

Yes:You can use complex numbers. So if $\theta$ is complex, then it can exceed $1$. For example, $\sin(1.57080 - 0.344701i) = 1.06$ (correct to 5dp at least).

Presumably you're at a stage where you aren't considering complex numbers, so the 'Yes' response here is a bit of a cheat. And I don't know of many cases where people go through explicit computations of $\sin(\theta)$ for complex $\theta$..

$\endgroup$ $\begingroup$

Recall the definition of the sine function:

As you can see, $c > a$, so $\sin(\alpha) \leq 1$.

$\endgroup$