Trigonometry Signs: A Quick Guide

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the sign of trigonometric functions, which determines whether the value is positive or negative.

This article will provide a quick guide to understanding how to determine the sign of trigonometric functions based on the angle’s location in the unit circle. We’ll also explore a simple trick to help you remember these signs.

The Unit Circle

The unit circle is a circle with a radius of 1 unit centered at the origin of a coordinate plane. It’s a powerful tool for visualizing trigonometric functions and understanding their relationships.

Each point on the unit circle corresponds to an angle in standard position, where the angle is measured counterclockwise from the positive x-axis. The coordinates of each point on the unit circle represent the cosine and sine of the angle, respectively.

Signs of Trigonometric Functions

The sign of trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) depends on the quadrant where the angle lies:

Quadrant	Sine (sin)	Cosine (cos)	Tangent (tan)
I	+	+	+
II	+	–	–
III	–	–	+
IV	–	+	–

Remembering the Signs: The CAST Rule

A handy mnemonic device called the CAST rule can help you remember the signs of trigonometric functions in each quadrant:

• Cosine is positive in Quadrant IV.
• All functions are positive in Quadrant I.
• Sine is positive in Quadrant II.
• Tangent is positive in Quadrant III.

You can visualize this by imagining a circle divided into four quadrants. Starting from the top right quadrant (Quadrant I) and moving counterclockwise, the CAST rule tells you which trigonometric function is positive in each quadrant.

Example

Let’s say you want to determine the sign of sin(150°). First, identify the quadrant where 150° lies. Since 150° is between 90° and 180°, it falls in Quadrant II. The CAST rule tells us that sine is positive in Quadrant II. Therefore, sin(150°) is positive.

Conclusion

Understanding the signs of trigonometric functions is crucial for solving trigonometry problems. The unit circle and the CAST rule provide simple ways to remember these signs and make solving trigonometric problems easier. By using these tools, you can accurately determine the sign of any trigonometric function based on the angle’s location.