How to Find Coterminal Angles Easily with Examples
Understanding coterminal angles is key for exams like CBSE, JEE, and competitive maths Olympiads. These angles help simplify problems in trigonometry and geometry. With a solid grasp of coterminal angles, students can solve tricky questions about rotations, unit circle, and trigonometric values more easily.
Formula Used in Coterminal Angles
The standard formula is: \( \theta_{\text{coterminal}} = \theta + 360^\circ n \) or \( \theta + 2\pi n \), where \( n \) is any integer. Add or subtract full rotations (in degrees or radians) to get all coterminal angles!
Here’s a helpful table to understand coterminal angles more clearly:
Coterminal Angles Table
| Angle (Degrees) | Is Coterminal With 45°? | Reason |
|---|---|---|
| -315° | Yes | -315° + 360° = 45° |
| 405° | Yes | 405° - 360° = 45° |
| 180° | No | Different terminal side |
| 765° | Yes | 765° - 2×360° = 45° |
This table shows how the pattern of coterminal angles appears in trigonometry problems. You can get any coterminal angle by adding or subtracting full rotations (360° or 2π) from a given angle.
Worked Example – Solving a Problem
1. The question: Find two coterminal angles for 120°.2. Step 1: Add 360° to 120°.
3. Step 2: Subtract 360° from 120°.
4. So, 480° and -240° are both coterminal angles with 120°.
5. Example in Radians: Find a positive coterminal angle for \(-\frac{\pi}{3}\).
6. Therefore, \(\frac{5\pi}{3}\) is a positive coterminal angle for \(-\frac{\pi}{3}\).
Practice Problems
- Find three coterminal angles for 75°.
- Is 400° coterminal with 40°?
- List all positive coterminal angles for –30° between 0° and 720°.
- Are \(\frac{\pi}{4}\) and \(-\frac{7\pi}{4}\) coterminal?
Common Mistakes to Avoid
- Confusing coterminal angles with reference angles — reference angles are always between 0° and 90°, but coterminal angles can be any angle at the same terminal side.
- Not checking for both positive and negative coterminal angles by forgetting to use both plus and minus multiples of 360° or 2π.
Real-World Applications
The concept of coterminal angles appears in fields like navigation (compass directions), engineering (rotating gears), and computer graphics (spinning objects). With Vedantu, students discover how these maths ideas connect to real-world situations and improve their problem-solving skills.
We explored the idea of coterminal angles, formulas, sample questions, and real-world links. Practising with Vedantu builds your understanding and makes you more confident for exams and daily problem-solving in maths.
If you want to review types of angles, try Angles and Its Types, or refresh the basics at Angle Definition. Need help with trigonometric functions for coterminal angles? Check Trigonometric Functions of Angles.
FAQs on What Are Coterminal Angles?
1. How do you find coterminal angles?
To find coterminal angles, add or subtract integer multiples of 360° (for degrees) or 2π (for radians) to the given angle. That is, coterminal angle = original angle ± 360° × n (where n is any integer). This method helps identify angles sharing the same terminal side on the unit circle.
2. What is a coterminal angle to 45°?
A coterminal angle to 45° can be found by adding or subtracting 360°. For example, 405° (45° + 360°) and -315° (45° - 360°) are both coterminal with 45°. These angles will have the same terminal side as 45° on the unit circle.
3. Are angles of 132° and −588° coterminal angles?
Yes, 132° and −588° are coterminal. If you add 2 × 360° (which is 720°) to −588°, you get 132° (−588° + 720° = 132°). This confirms that both angles share the same terminal side.
4. Are the angles 315° and −225° coterminal?
Yes, 315° and −225° are coterminal angles. Adding 360° to −225° gives 135°, but adding 360° twice results in 135° + 360° = 495°, which is not 315°. However, subtracting 360° from 315° results in −45°, not −225°, so these angles are not coterminal directly. Since 315° − (−225°) = 540°, which is 1.5 turns, these do not share the same terminal side and are not coterminal.
5. What are coterminal angles in radians?
In radians, coterminal angles are found by adding or subtracting multiples of 2π to the given angle. For example, θ and θ + 2πn (where n is any integer) are coterminal and share the same terminal side on the unit circle.
6. What is the definition of coterminal angles?
Coterminal angles are angles that share the same initial and terminal sides but may have different measures. They are formed by adding or subtracting full rotations (360° or 2π radians) to a given angle.
7. How do you find a negative coterminal angle?
To find a negative coterminal angle, subtract a multiple of 360° (or 2π radians) from the given angle so the result is negative. For example, to find a negative coterminal to 120°, subtract 360°: 120° − 360° = −240°. This angle is coterminal with 120°.
8. How can I use the unit circle to understand coterminal angles?
On the unit circle, coterminal angles end at the same point because they represent rotations ending at the same location. This visualization helps understand that coterminal angles share the same sine and cosine values.
9. What is the general formula for finding coterminal angles?
The formula for coterminal angles is: θ ± 360° × n (for degrees) or θ ± 2π × n (for radians), where θ is the original angle and n is any integer. This formula generates all possible coterminal angles to a given angle.
10. Can a reference angle also be a coterminal angle?
A reference angle is not usually coterminal with the original angle but is the smallest positive acute angle between the terminal side and the x-axis. However, coterminal angles can have the same reference angle when they share the same terminal side of the unit circle.
11. What are some examples of coterminal angles?
Examples of coterminal angles include:
• 30°, 390° (30° + 360°)
• –45°, 315° (–45° + 360°)
• π/6, 13π/6 (π/6 + 2π)
All these pairs share the same terminal side on the unit circle.
12. How can I practice finding coterminal angles?
You can practice coterminal angles by using worksheets or online calculators. Try finding coterminal angles for both positive and negative degree and radian measures, ensuring your results correspond to the same terminal side on the unit circle.
