in trigonometry when you have the ASTC quadrants. How do you work it out when the base angle is more than 90 degrees

In trigonometry, the **ASTC rule** helps determine the sign of trigonometric functions in the four quadrants. It stands for:

- **A**: All (All functions are positive in the first quadrant)
- **S**: Sine (Only sine is positive in the second quadrant)
- **T**: Tangent (Only tangent is positive in the third quadrant)
- **C**: Cosine (Only cosine is positive in the fourth quadrant)

### Handling Base Angles Greater Than 90°

When dealing with angles greater than 90 degrees, we first need to locate the correct quadrant and then apply the ASTC rule to determine the sign of the function. Here's a step-by-step guide:

#### 1. **Identify the quadrant** based on the angle:
- **Quadrant I**: \(0^\circ \leq \theta \leq 90^\circ\)
- **Quadrant II**: \(90^\circ < \theta \leq 180^\circ\)
- **Quadrant III**: \(180^\circ < \theta \leq 270^\circ\)
- **Quadrant IV**: \(270^\circ < \theta \leq 360^\circ\)

For angles larger than 90°, you'll need to locate the reference (or base) angle by reducing the angle to one in the range of \(0^\circ\) to \(90^\circ\).

#### 2. **Find the reference angle**:
- **Quadrant II**: The reference angle \( \alpha \) is \( 180^\circ - \theta \)
- **Quadrant III**: The reference angle \( \alpha \) is \( \theta - 180^\circ \)
- **Quadrant IV**: The reference angle \( \alpha \) is \( 360^\circ - \theta \)

The reference angle is always an acute angle (less than \(90^\circ\)) that helps you evaluate the trigonometric function for the original angle.

#### 3. **Determine the sign using ASTC**:
- In **Quadrant I** (A): All functions are positive.
- In **Quadrant II** (S): Only sine is positive, cosine and tangent are negative.
- In **Quadrant III** (T): Only tangent is positive, sine and cosine are negative.
- In **Quadrant IV** (C): Only cosine is positive, sine and tangent are negative.

#### Example:
If you are asked to find \( \sin(150^\circ) \):

1. The angle \(150^\circ\) is in **Quadrant II**.
2. The reference angle is \(180^\circ - 150^\circ = 30^\circ\).
3. In Quadrant II, sine is positive (ASTC rule).
4. So, \( \sin(150^\circ) = \sin(30^\circ) = 0.5 \).

### Summary:
- First, identify the quadrant.
- Then, calculate the reference angle.
- Finally, use the ASTC rule to apply the correct sign.

Would you like further clarification on any part of this? Here are some related questions that could deepen your understanding:

1. How do you calculate reference angles for angles larger than 360°?
2. How do you use the ASTC rule for negative angles?
3. How do reference angles work for radians instead of degrees?
4. How does ASTC help with evaluating inverse trigonometric functions?
5. How do you determine the exact values of common angles using ASTC?

**Tip**: Always remember that the reference angle is key to evaluating any trigonometric function beyond 90°.

In trigonometry when you have the ASTC quadrants, how do you work it out when the base angle is more than 90 degrees?

Learn how to apply the ASTC rule in trigonometry to work with angles greater than 90 degrees. This guide covers how to find reference angles and determine the signs of trigonometric functions in each quadrant, suitable for students in Grades 9-12.

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