Eli5: why sine of an angle or cos of an angle, can not be greater than unity?

In: Mathematics

Because you compare the length of two sides in a ratio. And for cos and sine, the denominator (compared side) is by definition the hypotenuse and that’s by definition the longest side. Hence that ratio can’t go over (or even reach) 1.

Tan can though.

* The values of sine and cosine are coordinates of points on the unit circle.

* Coordinates of a point are always less than or equal to (in magnitude) their distance from the origin.

* Points on the unit circle are exactly one unit from the origin.

Therefore, the values of sine and cosine must be less in magnitude than one.

Sine and Cosine are defined as the ratios of sides of a right-angle triangle.

The denominator in the ratio is the hypotenuse of the triangle and the numerator is one of the other sides of the same triangle.

The hypotenuse (denominator of the fraction) is always longer than the other sides of the right triangle, therefore their fractions are always smaller than 1.

Because that’s how they are defined based on the unit circle.

The unit circle is a circle with a radius of 1 and centered at (0,0). Sine and cosine are defined using points on the circle, which by definition can only have values between -1 and 1.