How do you calculate the exact value of sin 90°?

The exact value of sin 90° is 1.

To understand why sin 90° equals 1, we need to delve into the basics of trigonometry. The sine function, denoted as sin(θ), is a fundamental trigonometric function that relates the angle θ in a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. Specifically, for any angle θ, sin(θ) = opposite/hypotenuse.

When we consider the unit circle, which is a circle with a radius of 1 centred at the origin of a coordinate plane, the sine of an angle θ is the y-coordinate of the point where the terminal side of the angle intersects the circle. For an angle of 90°, the terminal side is a vertical line that intersects the unit circle at the point (0, 1). Here, the y-coordinate is 1, hence sin 90° = 1.

Another way to visualise this is by considering a right-angled triangle where one of the angles is 90°. In such a triangle, the side opposite the 90° angle is the hypotenuse itself. Since the hypotenuse is the longest side and is always equal to itself, the ratio of the opposite side to the hypotenuse (opposite/hypotenuse) is 1. Therefore, sin 90° = 1.

Understanding these concepts helps to see why the sine of 90° is exactly 1, reinforcing the relationship between angles and side lengths in trigonometry.