Coterminal Angles
In short: Coterminal angles are angles drawn from the same starting side that end on the very same terminal side, so they differ by a whole number of full turns of 360° or 2π radians.
Algebra 2TrigonometryF-TF.A.1
What is Coterminal Angles?
Coterminal angles are like a door swung open by 30° or by 390°: the measures differ, yet the door stops in the identical spot.
An angle \(\theta\) and every angle of the form \(\theta + 360^{\circ}k\), where \(k\) is an integer, are coterminal; in radians the form is \(\theta + 2\pi k\). Positive \(k\) adds counterclockwise turns and negative \(k\) adds clockwise ones. All of them have exactly the same sine, cosine and tangent.
How to spot it
- An angle is larger than 360° or negative, and you are asked where its terminal side lies.
- A question asks for one positive and one negative angle with the same terminal side as a given angle.
- A rotation count, like three and a half spins of a wheel, must be converted to a position.
- Two angles such as 11π/6 and -π/6 look different, and the task is to decide if they match.
The method
- Choose a whole number of turns \(k\) so that the angle moves into the window you want.
- Add \(360^{\circ}k\) in degrees, or \(2\pi k\) in radians, to the original angle.
- To find a standard angle, keep adding or subtracting full turns until it lies between 0° and 360°.
- To test a pair of angles, subtract them and see whether the difference is a multiple of a full turn.
Coterminal Angles: solved example
Problem
Find the angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(-835^{\circ}\).
Solution
Add full turns until the angle is positive. Three turns give \(3 \times 360^{\circ} = 1080^{\circ}\).
Then \(-835^{\circ} + 1080^{\circ} = 245^{\circ}\), which is in the required window. The coterminal angle is \(245^{\circ}\).
Try it
Question 1
Give one positive and one negative angle coterminal with \(130^{\circ}\).
Show the answer
Add a turn: \(130^{\circ} + 360^{\circ} = 490^{\circ}\). Subtract a turn: \(130^{\circ} – 360^{\circ} = -230^{\circ}\).
Question 2
Is \(\frac{11\pi}{6}\) coterminal with \(-\frac{\pi}{6}\)?
Show the answer
The difference is \(\frac{11\pi}{6} – \left(-\frac{\pi}{6}\right) = 2\pi\), one full turn, so yes.
Coterminal Angles vs. Reference Angle
Coterminal angles are a family of angles that share an exact terminal side, with measures differing by full turns. A reference angle is one acute number, measured from that side to the x-axis. Angles 40° and 400° are coterminal, but the reference angle of both is simply 40°.
Keep going
Lessons
Worksheets and quizzes
- Radians and the Unit Circle – Worksheet
- Radians and the Unit Circle – Quiz
- Radians and the Unit Circle – Worksheet
- Radians and the Unit Circle – Quiz
Related words
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