How to Find Coterminal Angles Using the Formula with Step by Step 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 Coterminal Angles Explained with Definition and Examples
1. What are coterminal angles?
Coterminal angles are angles in standard position that share the same initial and terminal sides but differ by a full rotation. They differ by multiples of 360° (in degrees) or 2π (in radians).
- If measured in degrees: θ + 360°k
- If measured in radians: θ + 2πk
- Where k is any integer (positive, negative, or zero)
2. How do you find coterminal angles?
To find coterminal angles, add or subtract multiples of 360° (or 2π) from the given angle. Follow these steps:
- Step 1: Identify the given angle θ.
- Step 2: Add or subtract 360° × k (or 2π × k in radians).
- Step 3: Choose any integer value of k.
3. What is the formula for coterminal angles?
The formula for coterminal angles is θ + 360°k (degrees) or θ + 2πk (radians), where k is any integer. This formula works because one full rotation equals 360° or 2π radians. For example, if θ = 60° and k = 2, then 60° + 720° = 780°, a coterminal angle.
4. How do you find a positive and a negative coterminal angle?
To find one positive and one negative coterminal angle, add 360° for a positive angle and subtract 360° for a negative angle. Example with 120°:
- Positive coterminal angle: 120° + 360° = 480°
- Negative coterminal angle: 120° − 360° = −240°
5. How do coterminal angles work in radians?
In radians, coterminal angles differ by multiples of 2π. The formula is θ + 2πk, where k is any integer.
- Example: For π/3, add 2π → π/3 + 2π = π/3 + 6π/3 = 7π/3
- Subtract 2π → π/3 − 2π = −5π/3
6. Do coterminal angles have the same trigonometric values?
Yes, coterminal angles have identical sine, cosine, and tangent values because they share the same terminal side on the unit circle. For example:
- sin(30°) = 1/2
- sin(390°) = 1/2
7. What is the smallest positive coterminal angle?
The smallest positive coterminal angle is the angle between 0° and 360° (or 0 and 2π radians). To find it:
- Keep adding or subtracting 360° until the angle is within the interval (0°, 360°).
- −450° + 360° = −90°
- −90° + 360° = 270°
8. What is the difference between coterminal and reference angles?
Coterminal angles share the same terminal side, while reference angles are the acute angles formed with the x-axis. Key differences:
- Coterminal angles differ by 360° or 2π.
- Reference angles are always between 0° and 90°.
9. Can coterminal angles be negative?
Yes, coterminal angles can be negative because you can subtract multiples of 360° (or 2π). Example:
- For 60°, subtract 360°
- 60° − 360° = −300°
10. Why are coterminal angles important in trigonometry?
Coterminal angles are important because they help simplify angles and evaluate trigonometric functions using familiar angles. They allow you to:
- Reduce large or negative angles to standard position (0°–360°).
- Use unit circle values easily.
- Solve equations like sin θ = 1/2 using general solutions θ = 30° + 360°k.
