What is the Value of Cos 360 Degrees and Why is Cos 360 = 1?
The concept of cos 360 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios, especially within trigonometry and geometry topics. Knowing the value and explanation of cos 360 degrees helps students tackle questions efficiently and avoid common mistakes in exams like JEE, NEET, and board tests.
What Is Cos 360?
Cos 360 means the value of the cosine function when the angle is 360 degrees. In trigonometry, cosine measures the ratio of the adjacent side to the hypotenuse in a right-angled triangle or the horizontal coordinate on the unit circle. Cos 360 appears in problems about the unit circle, angular rotations, periodicity, and standard trig tables. Understanding cos 360 also helps when relating degrees to radians and for memorizing standard trigonometric values.
Cos 360 Value Table
| Angle (°) | Angle (Radians) | cos θ | sin θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 1 | 0 | 0 |
| 90° | π/2 | 0 | 1 | ∞ |
| 180° | π | -1 | 0 | 0 |
| 270° | 3π/2 | 0 | -1 | ∞ |
| 360° | 2π | 1 | 0 | 0 |
Key Formula for Cos 360
Here’s the standard formula and reasoning:
\( \cos 360^\circ = \cos 0^\circ = 1 \)
Because cosine is periodic with a period of 360° or \( 2\pi \) radians, so:
\( \cos(\theta + 360^\circ) = \cos\theta \)
How to Calculate cos 360 in Radians
To convert 360° to radians:
\( 360^\circ \times \dfrac{\pi}{180^\circ} = 2\pi \) radians.
Therefore, \( \cos 360^\circ = \cos 2\pi = 1 \)
Unit Circle Explanation
The unit circle is a circle of radius 1 centered at the origin. Each angle θ has coordinates \( (\cos\theta, \sin\theta) \). After a full rotation (360°), the terminal side returns to (1, 0). So, \( \cos 360^\circ = 1 \).
Cos 360 Value: Direct Answer
The value of cos 360 degrees is 1. This is because, after one full revolution, you return to the starting point on the unit circle, so cos 360 = 1.
Examples & Applications
- Find cos 720°
cos 720° = cos(2 × 360°) = cos 0° = 1 - Find cos(360° + θ)
cos(360° + θ) = cos θ (by periodicity) - Solve: What is the value of cos²180° – sin²180°?
cos²180° – sin²180° = cos(2 × 180°) = cos 360° = 1
Comparison: Cos 0, 180, 270, 360
| Angle | Cosine Value |
|---|---|
| cos 0° | 1 |
| cos 180° | -1 |
| cos 270° | 0 |
| cos 360° | 1 |
Frequent Errors and Misunderstandings
- Mixing up radian and degree values.
- Confusing cos 360° with cos 180° (remember, cos 180° = -1, cos 360° = 1).
- Thinking sin 360° is 1; actually sin 360° is 0.
- Calculator rounding errors (sometimes cos 360° shows as 0.999999 or -1 due to decimal limitations — but the exact value is 1).
Relation to Other Concepts
The concept of cos 360 connects to the Trigonometric Values Table and the Unit Circle. It also uses periodicity, which is important in topics like Fourier analysis and wave functions.
Quick Reference: Standard Trigonometric Values
| Function | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| sin | 0 | 1 | 0 | -1 | 0 |
| cos | 1 | 0 | -1 | 0 | 1 |
| tan | 0 | ∞ | 0 | ∞ | 0 |
Classroom Tip
A quick way to remember cosine’s periodicity: "Cosine repeats every 360°." So, cos 0° = cos 360° = 1. Visualizing one full turn on the unit circle (starting and ending at (1, 0)) helps fix this in memory. Vedantu’s teachers often use this simple rotation image to make the learning process fun and clear in class.
Internal Links for Further Learning
We explored cos 360 from its definition, value, formula, unit circle meaning, stepwise calculations, error traps, and applications. Continue practicing with Vedantu to master trigonometric values and boost your confidence during exams and real-world problem-solving!
FAQs on Cos 360 Degrees (Value, Formula, Table & Meaning)
1. What is the value of cos 360°?
The value of cos 360° is 1. This is because 360° represents a complete revolution around the unit circle, returning to the starting point on the x-axis, where the cosine value is 1.
2. Why is cos 360° equal to 1?
Cos 360° equals 1 because an angle of 360° corresponds to one full rotation around the unit circle. The coordinates of the point on the unit circle at 360° are (1, 0), and the x-coordinate represents the cosine value.
3. How do you find cos 360° in radians?
To convert 360° to radians, multiply by π/180°: 360° × (π/180°) = 2π radians. Therefore, cos 360° = cos 2π = 1.
4. What is the formula for cos(360°)?
The cosine function is periodic with a period of 360° (or 2π radians). Therefore, cos(360°) = cos(0°) = 1. More generally, cos(x + 360n°) = cos(x) for any integer n.
5. What is the value of sin 360°?
The value of sin 360° is 0. Similar to cosine, one full rotation around the unit circle at 360° lands on the point (1,0). The y-coordinate represents the sine value.
6. What is cos (360° + θ)?
Due to the periodicity of the cosine function, cos (360° + θ) = cos θ. Adding or subtracting multiples of 360° does not change the cosine value.
7. Is cos 360° the same as cos 0°?
Yes, cos 360° = cos 0° = 1. Both angles represent the same point on the unit circle.
8. How does periodicity affect cosine values at multiples of 360°?
The periodicity of the cosine function means that cos(360n°) = 1 for any integer n. This is because each multiple of 360° represents a complete revolution around the unit circle, resulting in the same cosine value (1).
9. What is cos(-360°)?
Because cosine is an even function, cos(-360°) = cos(360°) = 1. The cosine of a negative angle is equal to the cosine of the corresponding positive angle.
10. What are the values of cos 0°, cos 90°, cos 180°, cos 270°, and cos 360°?
Here's a quick summary of the cosine values at these key angles:
- cos 0° = 1
- cos 90° = 0
- cos 180° = -1
- cos 270° = 0
- cos 360° = 1
11. How can I remember the trigonometric values for standard angles?
Use a unit circle diagram and memorize the key values for 0°, 30°, 45°, 60°, and 90°. Understanding the symmetry of the unit circle helps extend these values to other angles. Practice problems and create memory aids, like mnemonics or diagrams.
