知识点 6

Use the unit circle or special-angle reasoning to connect angles and coordinates (SAT).

On the unit circle, coordinates come from cosine and sine. Special angles and quadrant signs give the exact value.

核心知识

On the unit circle, the point at angle 𝜃 has coordinates (cos⁡𝜃,sin⁡𝜃). Special angles (30°, 45°, 60° and their multiples) give exact coordinate values.

深入理解

Rule: On the unit circle, coordinates are (cos⁡𝜃,sin⁡𝜃).

  • Start with the reference angle.
  • Use the exact special-angle coordinates.
  • Then fix the signs for the quadrant.

Cosine is the x-coordinate, sine is the y-coordinate.

分步讲解

  1. Find the reference angle (the acute angle the terminal side makes with the x-axis).
  2. Look up or recall the coordinates for that reference angle on the unit circle.
  3. Adjust signs based on the quadrant: x is negative in Q2 and Q3, y is negative in Q3 and Q4.
  4. Read off cos⁡𝜃(𝑥 −coordinate) or sin⁡𝜃 (y-coordinate) as needed.

常见误解

  • Swapping sine and cosine — cosine is the x-coordinate, sine is the y-coordinate.
  • Forgetting to adjust signs for the quadrant.
  • Using degree values in radian formulas (or vice versa) without converting.
题目

示例解析

What is the value of cos⁡2𝜋3?

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