Table of Contents

Last modified on August 3rd, 2023

A reference angle is the smallest acute angle that the terminal arm of the angle makes with the x-axis when drawn on a coordinate plane. Regardless of the location of the terminal side, the reference angle measures the closest distance of that side from the x-axis.

The reference angle is used for simplifying the calculations related to trigonometric functions with different angles. Their value is always between 0 and 90° when measured in degrees or 0 and π/2 when measured in radians. A reference angle always uses an x-axis as its frame of reference.

Looking at the picture above, every angle is measured from the positive part of the x-axis to its terminal side by traveling in a counterclockwise direction.

The two axes, x, and y divide the plane into four quadrants, named I, II, III, and IV. The numbering starts from the upper right one, the first quadrant, where both coordinates are positive as we continue to move in the anticlockwise direction.

Normally, the four trigonometric functions: sine, cosine, tangent, and cotangent give the same value for an angle and its reference angle. The only thing that varies is the sign.

Follow the mnemonic rule: ‘**A**dd **S**ugar **T**o **C**offee’ or ‘**A**ll **S**tudents **T**ake **C**alculus’ to remember when these functions have positive values.

**‘A’** for all: In the first quadrant, all functions have positive value

**‘S’** for sine: In the second quadrant, only the sine function has positive value

**‘T’** for tangent: In the third quadrant, tangent and cotangent have positive values

**‘C’** for cosine: In the fourth quadrant, only the cosine function has positive value

We can find reference angles depending on which quadrant the terminal side of the angle is located in either degrees or radians. There are four possible cases:

**1) When Calculated in Degrees**

**Case 1**: (For Angles between 0° to 90°) – When the terminal side is on the first quadrant, the reference angle is the same as the given angle. So, if the given angle is 45°, then its reference angle is also 45°.

Hence,

Reference Angle = Given Angle

**Case 2**: (For Angles between 90° to 180°) – When the terminal side is on the second quadrant, the reference angle is 180° minus our given angle. So, if the given angle is 105°, then its reference angle is (180° – 105° = 75°).

Hence,

Reference Angle = 180° – Given Angle

**Case 3**: (For Angles between 180° to 270°) – When the terminal side is on the third quadrant, the reference angle is the given angle minus 180°. So, if the given angle is 200°, then its reference angle is (200° – 180° = 20°).

Hence,

Reference Angle = Given Angle – 180°

**Case 4**: (For Angles between 270° to 360°) – When the terminal side is on the fourth quadrant, the reference angle is 360° minus our given angle. So, if the given angle is 310°, then its reference angle is (360° – 310° = 50°).

Hence,

Reference Angle = 360° – Given Angle

**2) When Calculated in Radians**

When calculated in radians: 180° = π, 360° = 2π, 270 = 2π/2, and 90° = π/2

Thus, the formulas become:

**Case 1**: (For Angles between 0° to 90°) – First quadrant

Reference Angle = Given Angle

**Case 2**: (For Angles between 90° to 180°) – Second quadrant

Reference Angle = π – Given Angle

**Case 3**: (For Angles between 180° to 270°) – Third quadrant

Reference Angle = Given Angle – π

**Case 4**: (For Angles between 270° to 360°) – Fourth quadrant

Reference Angle = 2 π – Given Angle

Given below is a table showing the values of reference angles in radians and degrees in the four coordinate planes. Assume that the original angle is

Quadrant | Reference Angle (in radians) | Reference Angle (in degrees) |

I | θ | θ |

II | π – θ | 180° – θ |

III | θ – π | θ – 180° |

IV | 2π – θ | 360° – θ |

1) Used to find the value of an angle with trigonometric functions such as the sine or cosine value of any angle. This is done using corresponding reference angle in the first quarter as discussed above in the ‘Reference Angles and Trigonometric Functions’ section.

2) Used in preparing unit circles, having a radius of one, that helps to represent reference angles in various trigonometric functions. The sine function is the y-coordinate on the unit circle and cosine function is the x-coordinate on the unit circle.

Given below are two unit circles one showing the values in degrees and the other in radians.

The degrees unit circle and the radians unit circle have the same angles on them. Notice that there are a selected number of reference angles shown in the unit circles since they have their in simple decimal numbers.

**Determine the reference angle corresponding to each of the given angles.****a) 140°, b) 260°, c) 65°, d) 340°**

Solution:

a) This lies in the second quadrant,

Thus,

Reference Angle = 140° – Given Angle, here given angle = 140°

= 180° – 140°

= 40°

b) 260° lies in the third quadrant,

Thus,

Reference Angle = Given Angle – 180°, here given angle = 260°

= 260° – 180°

= 80°

c) 65° lies in the first quadrant,

Thus,

Reference Angle = Given Angle, here given angle = 65°

= 65°

d) 340° lies in the fourth quadrant,

Reference Angle = 360° – Given Angle, here given angle = 340°

= 360° – 340°

= 20°

**Find the reference angle for 16 π/9 radians.**

Solution:

As the degree is given in radians, we need to think from 0 radians to 2 π radians for the positive x-axis, and π radians for the negative x-axis.

Converting the fractional form to the decimal form, we get,

16 π/9 radians = 1.77 π radians

Since, 1 cycle is 2π radians, so it is a bit less than two cycles but more than 3/2 = 1.5. This angle thus lies in the fourth quadrant between 3π/2 and 2π radians.

Now, as we know, for fourth quadrant

Reference Angle = 2 π – Given Angle, here given angle = 16/9 π radians

= 2 π – 16/9 π

= 2/9 π

Hence the reference angle to 16/9 π radians is 2/9 π.

Last modified on August 3rd, 2023

Very Nice Material for Trigonometry