Sign of Trigonometric Functions

Understanding the sign (positive or negative) of trigonometric functions in different quadrants is crucial for solving equations, graphing functions, and analyzing angles. Here's a breakdown of the sign of trigonometric functions in each quadrant:

1. Quadrant I (0° to 90° or 0 to π2):

  • Sine (sin): Positive
  • Cosine (cos): Positive
  • Tangent (tan): Positive

In the first quadrant, all trigonometric functions are positive because both the x-coordinate and y-coordinate are positive.

2. Quadrant II (90° to 180° or π2 to π):

  • Sine (sin): Positive
  • Cosine (cos): Negative
  • Tangent (tan): Negative

In the second quadrant, only the sine function is positive because the y-coordinate is positive. The cosine and tangent functions are negative as the x-coordinate is negative.

3. Quadrant III (180° to 270° or π to 3π2):

  • Sine (sin): Negative
  • Cosine (cos): Negative
  • Tangent (tan): Positive

In the third quadrant, both the sine and cosine functions are negative because both the x-coordinate and y-coordinate are negative. The tangent function is positive because the ratio of negative over negative is positive.

4. Quadrant IV (270° to 360° or 3π2 to 2π):

  • Sine (sin): Negative
  • Cosine (cos): Positive
  • Tangent (tan): Negative

In the fourth quadrant, the sine function is negative because the y-coordinate is negative. The cosine function is positive because the x-coordinate is positive. The tangent function is negative as the ratio of negative over positive is negative.

5. Key Takeaways:

  • Sine Function:
    • Positive in Quadrants I and II.
    • Negative in Quadrants III and IV.
  • Cosine Function:
    • Positive in Quadrants I and IV.
    • Negative in Quadrants II and III.
  • Tangent Function:
    • Positive in Quadrants I and III.
    • Negative in Quadrants II and IV.

6. All Students Take Calculus (ASTC) Rule:

  • A helpful acronym to remember the sign of trigonometric functions in each quadrant.
  • All: All functions are positive in Quadrant I.
  • Students: Sine is positive in Quadrant II.
  • Take: Tangent is positive in Quadrant III.
  • Calculus: Cosine is positive in Quadrant IV.