How to Use the CAST Circle?

The CAST rule is a mnemonic used in trigonometry to remember which trigonometric functions (sine, cosine, and tangent) are positive in each of the four quadrants of the Cartesian coordinate system or the unit circle. It's often visualized on a circle, hence "CAST circle".

Understanding the CAST rule helps determine the sign (+ or -) of sin(θ), cos(θ), and tan(θ) for any angle θ, without needing to calculate the exact value.

What is the CAST Rule?

The CAST rule is applied starting from Quadrant 4 and moving counter-clockwise:

• C in Quadrant 4: Only Cosine is positive.
• A in Quadrant 1: All (Sine, Cosine, and Tangent) are positive.
• S in Quadrant 2: Only Sine is positive.
• T in Quadrant 3: Only Tangent is positive.

This corresponds directly to the information provided:

1. C in quadrant 4 means only cosine is positive in quadrant 4.
2. A in quadrant 1 means all are positive (sine, cosine, and tangent) in quadrant 1.
3. S in quadrant 2 means only sine is positive in quadrant 2.
4. T in quadrant 3 means only tangent is positive in quadrant 3.

Applying the CAST Rule Step-by-Step

To use the CAST rule for a given angle, follow these steps:

1. Identify the Quadrant: Determine which quadrant the angle falls into. Angles are typically measured counter-clockwise from the positive x-axis.
   • Quadrant 1: 0° to 90° (or 0 to π/2 radians)
   • Quadrant 2: 90° to 180° (or π/2 to π radians)
   • Quadrant 3: 180° to 270° (or π to 3π/2 radians)
   • Quadrant 4: 270° to 360° (or 3π/2 to 2π radians)
     (Note: Angles greater than 360° or less than 0° are handled by finding their coterminal angle within 0° to 360°).
2. Apply the CAST Rule: Based on the quadrant identified, use the CAST mnemonic to determine which trigonometric function(s) are positive in that quadrant.
   • If the function you are evaluating is the one indicated as positive for that quadrant, its value will be positive.
   • If it is not the function indicated as positive (and it's not one of the 'All' in Quadrant 1), its value will be negative.

Sign Summary by Quadrant

Here's a table summarizing the signs of the basic trigonometric functions in each quadrant according to the CAST rule:

Quadrant	Angles (Degrees)	Angles (Radians)	Positive Functions	Negative Functions	CAST Letter
Q1	0° to 90°	0 to π/2	Sin, Cos, Tan	None	A (All)
Q2	90° to 180°	π/2 to π	Sin	Cos, Tan	S (Sine)
Q3	180° to 270°	π to 3π/2	Tan	Sin, Cos	T (Tangent)
Q4	270° to 360°	3π/2 to 2π	Cos	Sin, Tan	C (Cosine)

Practical Example

Let's determine the signs of sin(210°), cos(210°), and tan(210°).

1. Identify the Quadrant: 210° is between 180° and 270°. This falls into Quadrant 3.
2. Apply the CAST Rule: Quadrant 3 corresponds to 'T' in CAST. This means only Tangent is positive in Quadrant 3.
   • Therefore, tan(210°) will be positive.
   • Since Sine and Cosine are not positive in Quadrant 3, sin(210°) and cos(210°) will both be negative.

The CAST rule is a fundamental tool for understanding trigonometric relationships and solving problems involving angles outside the first quadrant.