Question

Why cant the value of sine never exceed one

Answer 1

Have at it, Kai.

Answer 2

If you remember SOH CAH TOA, this just means that sine is the ratio of the length of the opposite side to the length of the hypotenuse to the angle. Now notice if you change the angle by rotating the hypotenuse around like a circle, then you can see that the opposite and adjacent sides will have to change lengths to keep a right angle.

So if we make the angle 90 degrees, we'll basically have a triangle that's got two sides of equal length and the other side with length 0. You can't ever rotate yourself into the other side being longer.

Maybe that was kind of confusing, anyone else who wants to draw more pictures to help explain it might be more helpful than me, I gotta run!

Answer 3

i agree with @Kainui

Answer 4

To add to @Kainui's post, consider the fact that the hypotenuse will be the longest segment in any right triangle. Since sine is a ratio of the length of the leg opposite a certain angle to the length of the hypotenuse \(\bigg(\text{i.e. }\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\bigg)\).

The numerator of this ratio will never exceed the size of the denominator, but it's possible for \(\text{opp}\) to have the same magnitude as \(\text{hyp}\), so you will always have a fraction less than 1 (for the first and second quadrants).

For a visual aid:

As you go around the circle in this fashion, you'll see what Kainui was referring to.