# 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.