Read exact sine values on the unit circle
The idea
Measure angles counterclockwise from the positive x-axis, using radians.
Step by step
- Measure angles counterclockwise from the positive x-axis, using radians.
- Recall that the unit-circle point at angle θ is (cos θ, sin θ). Sine is its vertical coordinate.
- Use the standard sine values: sin(0) = 0, sin(π/6) = , sin(π/2) = 1, sin(π) = 0 and sin(3π/2) = −1. The π/6 value follows from a 30°–60°–90° triangle with hypotenuse 1.
- At 7π/6, the reference angle is π/6, but the point lies in the third quadrant below the x-axis. Its y-coordinate is , so sin(7π/6) = −1/2.
Example
An angle measures 0 radians. What is its exact sine?
Answer
0
On the unit circle, sine is the vertical coordinate. At 0 radians that coordinate is 0, so sin(0) = 0.
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Mark the angles 0, π/2, π and 3π/2 on a unit circle. Explain their sine values using vertical coordinates.
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