Understanding Why the Values of Sine and Cosine Functions Are Between -1 and 1

The values of the sine (sin) and cosine (cos) functions are constrained between -1 and 1 due to their definitions in the context of the unit circle in trigonometry. This limitation is rooted in the fundamental principles of trigonometry and the geometry of the unit circle.

Unit Circle Definition

In trigonometry, the sine and cosine of an angle are defined as the y-coordinate and x-coordinate, respectively, of a point on the unit circle—a circle with a radius of 1 corresponding to that angle. The equation of the unit circle is given by:

x2 y2 1

Coordinates on the Unit Circle

For any angle θ, the coordinates of the point on the unit circle can be expressed as:

x cos(θ)

y sin(θ)

Since every point on the unit circle satisfies the equation x2 y2 1, both cos(θ) and sin(θ) must lie within the range of values that keep this equation valid. This means:

-1 ≤ cos(θ) ≤ 1

-1 ≤ sin(θ) ≤ 1

Right Triangle Insight

A right triangle cannot have one side longer than the hypotenuse, which is opposite the right angle. Since the sine and cosine are ratios of a non-hypotenuse side to the hypotenuse, the fraction formed cannot be greater than 1. This is further explained through a simple right-angled triangle:

sin(θ) perpendicular / hypotenuse

cos(θ) base / hypotenuse

Since the hypotenuse is the longest side of the triangle, both sine and cosine can never be larger than 1.

Further Clarification

The simplest way to grasp this concept is by considering a unit circle, where the radius is 1. Both sine and cosine are percentages of the radius in a trigonometric circle. Therefore, the maximum value for either sine or cosine is the radius, which is 1.

In summary, the values of sine and cosine are always between -1 and 1 due to their definitions in the context of the unit circle and the geometric properties of right triangles. This is a fundamental principle in trigonometry that underpins much of its theoretical and practical applications.