why sin(3π/2) = -1
The sine function is defined as the y-coordinate of the point on the unit circle that corresponds to an angle in standard position.
In the unit circle, the angle 3π/2 (270 degrees) is measured counterclockwise from the positive x-axis and intersects the negative y-axis. The y-coordinate of the intersection point is -1.
Therefore, sin(3π/2) is equal to -1.
To explain why sin(3π/2) = -1, let's break it down step-by-step:
Step 1: Understand the unit circle
The unit circle is a circle with a radius of 1, centered at the origin (0,0) in a coordinate plane. It is often used to understand and calculate trigonometric functions.
Step 2: Representing angles on the unit circle
Angles on the unit circle are measured in radians.
- One complete revolution around the unit circle (360 degrees) is equal to 2π radians.
- The coordinate on the unit circle at any given angle (in radians) is represented by the coordinates (x, y), where x is the cosine value and y is the sine value.
Step 3: Understanding the angle 3π/2
3π/2 is an angle measured in radians. It is equivalent to 270 degrees.
Step 4:Finding the sine value
To find the sine value of the angle 3π/2, we need to find the y-coordinate on the unit circle at this angle.
Step 5: Locate the angle 3π/2 on the unit circle
To locate the angle 3π/2 on the unit circle, start from the positive x-axis (right side) and rotate counterclockwise.
At this angle, the point on the unit circle will be at the coordinate (0, -1).
Step 6: Determine the sine value
The sine value is equal to the y-coordinate. In this case, the y-coordinate is -1. Therefore, sin(3π/2) = -1.
So, the angle 3π/2 corresponds to the coordinate (0, -1) on the unit circle, resulting in a sine value of -1.