Question

Does this make sense? Easy medal!!
In response to this question:
You know that for any , neither sin nor cos can be greater than 1. How can you explain this using the unit circle definitions of sine and cosine? How can you explain it using the right triangle definitions of sine and cosine?
As a follow-up question, consider why it is important to have both the right triangle definitions of sine and cosine and the unit circle definitions of sine and cosine.

My answer will be in the comments:

Answer 1

1. In a unit circle, the hypotenuse of any triangle is one, because the radius of the circle is one. Any triangle with the angle theta has the point on the circle of (x,y), or (cos, sin). cos (theta)^2+sin(theta)^2 has to be equal to 1 because that is the definition of a point that is on the unit circle. So, there would never be able to have cos and sin values that are greater than one, because even if the other value was negative, they are still squared, resulting in a positive number. In that case, there would have to be an imaginary numbers and I don't believe that is possible on a unit circle. On a triangle, the hypotenuse is always the largest side, and a^2+b^2=c^2. The same principle applies as in the unit circle.
It is important to have these definitions because sometimes one may forget one of these definitions, but is able to remember the other. Through this, they are able to derive the basic functions of a unit circle or a right triangle, and therefore able to solve the problem.

Answer 2

Seems fine.

Answer 3

Thanks! @Zelda