How do you determine 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 and the unit circle. The sine function relates to the y-coordinate of a point on the unit circle, which is a circle with a radius of 1 centred at the origin of a coordinate plane. When we measure angles in standard position (starting from the positive x-axis and moving counterclockwise), an angle of 90° corresponds to a point directly above the origin on the unit circle.

At 90°, the coordinates of this point are (0, 1). The sine of an angle in the unit circle is defined as the y-coordinate of the corresponding point. Therefore, for 90°, the y-coordinate is 1, making sin 90° equal to 1.

Another way to understand this is by considering a right-angled triangle. If you imagine a right-angled triangle where one of the angles is 90°, the side opposite this angle is the hypotenuse. The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the hypotenuse. For a 90° angle, the opposite side is the hypotenuse itself, so the ratio is 1 (since any number divided by itself is 1).

Thus, both the unit circle and right-angled triangle perspectives confirm that the exact value of sin 90° is 1.