Why, in the real number system, is sinΘ and cosΘ never more than 1?

Answer 1

Because of (one of) the definition(s) of sin and cos.
Which is: take a unit circle, and draw in any radius at some angle theta. Then sin theta = the y value, and cos theta = the x value. Those are obviously less than 1.

Or you could just take a typical right angled triangle - the length of the opposite/adjacent sides are always less than the length of the hypotenuse.

Answer 2

Given that the x and y coordinates of x² + y² = r² are:

x = r cosθ and y = r sinθ

The domain for the circle is -r ≤ x ≤ r
and the range is -r ≤ y ≤ r

So from the domain: -r ≤ r cosθ ≤ r ie -1 ≤ cosθ ≤ 1
and from the range -r ≤ r sinθ ≤ r ie -1 ≤ sinθ ≤ 1

Answer 3

Because, in a right triangle, the sine is the ratio of the opposite to the hypotenuse, and the cosine is the ratio of the adjacent to the hypotenuse. Once either the opposite or the adjacent becomes the same length as the hypotenuse, the triangle "collapses" into a line.

It's therefore impossible for either the opposite or the adjacent to be any larger than the hypotenuse, so the ratio of the length of either to the length of the hypotenuse can be at most 1.

Answer 4

The sin() and cos() of an angle are measured (in a right triangle) by taking the opposite or adjacent sides of the triangle and dividing them by the length of the hypotenuse. Since the hypotenuse is the longest side of the triangle, the answer can never be greater than 1

Answer 5

sine=opposite over hypotenuse
cosine=adjacent over hypotenuse.

the hypotenuse is the longest side of a right triangle, therefore both must be <=1 (& >=-1)

Answer 6

sin and cos are fractions. by definition, a fraction is less than one