How to Name Trigonometric Functions Without A Calculator

Naming trigonometric functions without a calculator requires understanding standard notation and reference angles. This guide explains how to properly name sine, cosine, and tangent functions for any angle, including those not commonly memorized.

Standard Notation for Trigonometric Functions

The standard notation for trigonometric functions is based on the unit circle, where each angle corresponds to a point (x, y) on the circle. The basic functions are:

sin(θ) = y-coordinate of the point on the unit circle

cos(θ) = x-coordinate of the point on the unit circle

tan(θ) = y/x (ratio of y to x coordinates)

For angles outside the first quadrant (0° to 90°), we use reference angles to determine the function names. The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis.

Using Reference Angles

To find the reference angle for any angle θ:

Determine the quadrant of θ
Subtract the appropriate multiple of 90° to find the reference angle

For example, for θ = 120° (second quadrant):

Reference angle = 180° - 120° = 60°

The sign of each trigonometric function depends on the quadrant:

Quadrant	sin(θ)	cos(θ)	tan(θ)
I (0°-90°)	Positive	Positive	Positive
II (90°-180°)	Positive	Negative	Negative
III (180°-270°)	Negative	Negative	Positive
IV (270°-360°)	Negative	Positive	Negative

Special Angles and Their Names

Some angles have special names based on their relationship to the unit circle:

30° (π/6 radians)
45° (π/4 radians)
60° (π/3 radians)
90° (π/2 radians)
180° (π radians)
270° (3π/2 radians)
360° (2π radians)

The names of these angles are based on their position on the unit circle and their relationship to the x and y axes.

Examples of Naming Trigonometric Functions

Let's look at several examples to demonstrate how to name trigonometric functions without a calculator:

Example 1: 120° Angle

120° is in the second quadrant. The reference angle is 60°.

Using the quadrant signs:

sin(120°) = sin(60°) = √3/2 (positive)
cos(120°) = -cos(60°) = -1/2 (negative)
tan(120°) = -tan(60°) = -√3 (negative)

Example 2: 210° Angle

210° is in the third quadrant. The reference angle is 30°.

Using the quadrant signs:

sin(210°) = -sin(30°) = -1/2 (negative)
cos(210°) = -cos(30°) = -√3/2 (negative)
tan(210°) = tan(30°) = √3/3 (positive)

Example 3: 300° Angle

300° is in the fourth quadrant. The reference angle is 60°.

Using the quadrant signs:

sin(300°) = -sin(60°) = -√3/2 (negative)
cos(300°) = cos(60°) = 1/2 (positive)
tan(300°) = -tan(60°) = -√3 (negative)

Reference Table of Trigonometric Function Names

This table shows the names of trigonometric functions for common angles:

Angle	sin(θ)	cos(θ)	tan(θ)
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	Undefined
120°	√3/2	-1/2	-√3
180°	0	-1	0
270°	-1	0	Undefined
360°	0	1	0

FAQ

What is the standard notation for trigonometric functions?

The standard notation uses sin(θ) for the y-coordinate, cos(θ) for the x-coordinate, and tan(θ) for the ratio of y to x on the unit circle.

How do I find the reference angle for any angle?

Subtract the appropriate multiple of 90° based on the quadrant of the angle. For example, for 120° in the second quadrant, subtract 90° to get 30°.

What are the signs of trigonometric functions in each quadrant?

In the first quadrant (0°-90°), all functions are positive. In the second quadrant (90°-180°), sine is positive while cosine and tangent are negative. In the third quadrant (180°-270°), tangent is positive while sine and cosine are negative. In the fourth quadrant (270°-360°), cosine is positive while sine and tangent are negative.

What are the names of special angles in trigonometry?

Special angles include 30°, 45°, 60°, 90°, 180°, 270°, and 360°. These angles have specific names based on their position on the unit circle.

How do I name trigonometric functions for angles outside the first quadrant?

First find the reference angle, then apply the appropriate sign based on the quadrant. For example, for 210° in the third quadrant, the reference angle is 30°, and sine is negative while cosine and tangent are positive.