How to find Coterminal angles? Trigonometry is a subject that involves the observation of evaluation or analysis of angles. It includes trigonometric functions and trigonometric operations for the determination of unknown angles. Trigonometry also has formulas for different operations and has standard trigonometric values or ratios under different angles for the basic functions sine, cosine, tangent, cotangent, secant, and cosecant.
The given article focuses on the sub-topic of trigonometry coterminal angles. The content of the article includes a brief description of coterminal angles and their types, the formula of coterminal angles, and the method to find them. Some sample problems are also included to understand the method of calculation.
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Coterminal Angles
Coterminal angles can be defined as the angles having the same initial and terminal sides. Coterminal angles hold a standard position in each quadrant which determines their different values. When we see coterminal angles in trigonometry, the coterminal angles have the same values for the functions of sine, cosine, and tangent. These angles are generally determined by the mathematical operation of adding or subtracting 360 degrees or 2π to the given angle.
While determining a coterminal angle the angles move clockwise or anticlockwise they will coincide at the same terminal side. As per their rotation, coterminal angles can be positive or negative.
- Positive coterminal angle
When the rotation is anticlockwise and the value of ‘n’ is found to be positive it is considered to be the positive coterminal angle.
In θ ± 360n, the n attends positive value when the rotation is anticlockwise.
- Negative coterminal angle
When the rotation is clockwise and the value of ‘n’ is found to be negative it is considered to be the negative coterminal angle.
In θ ± 360n, the n attends negative value when the rotation is clockwise.
How to Find Coterminal Angles?
The coterminal angles are determined by the derived coterminal angles formula that uses ‘θ’ as a reference for the operation. Hence, the value of θ is required to find coterminal angles whether in degree or radian.
The mathematical formula of coterminal angles is,
- In Degrees
θ ± 360n
- In Radian
θ±2πn
Where,
n is the integer
As studied earlier it is known that coterminal angles can be determined in degrees or radians. And, the 360n or 2πn are the multiples of the given integer. Therefore,
- To determine the coterminal angle in degrees, add or subtract multiples of 360 to the given angle.
- To determine the coterminal angles in radians, add or subtract multiples of 2π to the given angle.
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Sample Problems on How to Find Coterminal angles?
Question 1: Find the coterminal angle of π/2.
Solution:
Given:
The angle is θ = π/2. (In radians)
Now,
Add or subtract multiples of 2π from the angle,
Let’s subtract 2π from the given angle.
=> π/2 – 2π
=> -3π/2
Hence, the coterminal angle of π/2 is -3π/2.
Question 2: Find the coterminal angle of π/4.
Solution:
Given:
The angle is θ = π/4 (In radians)
Now,
Add or subtract multiples of 2π from the angle,
Let’s add 2π from the given angle.
=> π/4 + 2π
=> 9π/4
Hence, the coterminal angle of π/4 is 9π/4.
Question 3: Find the coterminal angle of π/6.
Solution:
Given
The angle is θ = π/6 (In radians)
Now,
Add or subtract multiples of 2π from the angle,
Let’s subtract 2π from the given angle.
=> π/6 – 2π
=> -11π/6
Hence, the coterminal angle of π/6 is -11π/6.
Question 4: Find the coterminal angles of 30°.
Solution:
Given:
The angle θ = 30°
For anticlockwise, let n = 1
=> θ + 360n
=> 30 + 360(1)
=> 390°
For clockwise, let n = -2
=> θ – 360n
=> 30 – 360(-2)
=> -690°
Question 5: Find the coterminal angles of 40°.
Solution:
Given:
The angle θ = 40°
For anticlockwise, let n = 1
=> θ + 360n
=> 40 + 360(1)
=> 400°
For clockwise, let n = -2
=> θ – 360n
=> 40 – 360(-2)
=> 40 – 720
=> -680°
Question 6: Find the coterminal angles of -450°.
Solution:
Given:
The angle θ = -450°
For anticlockwise, let n = 1
=> θ + 360n
=> -450 + 360(1)
=> -90°
For clockwise, let n = -2
=> θ – 360n
=> -450 – 360(-2)
=> -450 – 720
=> -1170°
Practice Problems on How to find Coterminal angles?
1. Find three coterminal angles to 𝜃 = 30∘ within the range 0∘ to 360∘.
2. Determine one positive and one negative coterminal angle to 𝛼=−120∘.
3. Given β=450∘, find two coterminal angles in the range 0∘ to 360∘.
4. Find three coterminal angles to 𝛾=−210∘ within the range −360∘ to 0∘.
5. Determine two coterminal angles to 𝛿=720∘ within the range 0∘ to 360∘.