# 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 $\frac{\pi }{2}$):

• Sine ($\mathrm{sin}$): Positive
• Cosine ($\mathrm{cos}$): Positive
• Tangent ($\mathrm{tan}$): 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 $\frac{\pi }{2}$ to $\pi$):

• Sine ($\mathrm{sin}$): Positive
• Cosine ($\mathrm{cos}$): Negative
• Tangent ($\mathrm{tan}$): 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 $\pi$ to $\frac{3\pi }{2}$):

• Sine ($\mathrm{sin}$): Negative
• Cosine ($\mathrm{cos}$): Negative
• Tangent ($\mathrm{tan}$): 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 $\frac{3\pi }{2}$ to $2\pi$):

• Sine ($\mathrm{sin}$): Negative
• Cosine ($\mathrm{cos}$): Positive
• Tangent ($\mathrm{tan}$): 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.