Why is Cos 180 Degrees Equal to -1? (Unit Circle Explained)
The concept of Value of cos 180 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding this trigonometric value helps in solving triangle problems, working with the unit circle, and answering quick MCQs in competitive exams. Cos 180 degrees is a classic formula-based value that every Indian school student—whether in CBSE, ICSE, or JEE prep—needs to remember.
What Is Value of cos 180?
Value of cos 180 means "what is the cosine of a 180-degree angle?" A cosine value tells us how far left or right a point is on the unit circle as you sweep an angle from the positive x-axis. Specifically, cos 180° equals -1. You’ll find this concept applied in areas such as geometry (angles in polygons or straight lines), algebraic transformations, and physics (wave phases).
Key Formula for Value of cos 180
Here’s the standard formula: \( \cos 180^\circ = -1 \)
Quick Reference Table: Common Cosine Values
| Angle (°) | 0 | 90 | 180 | 270 | 360 |
|---|---|---|---|---|---|
| cos θ | 1 | 0 | -1 | 0 | 1 |
Why Is cos 180 Negative? (Unit Circle Explanation)
On the unit circle, cos 180° is the point where the terminal side of the angle lies on the negative x-axis. Since cosine measures the horizontal distance from the origin, at 180° you are one unit to the left—so cos 180° = -1. In Quadrant II (90° to 180°), cosine is always negative. This also explains why cos 0 is positive, but cos 180 is negative.
Cos(180° – θ) Formula & Examples
For any angle θ, the formula is: \( \cos(180^\circ - \theta) = -\cos\theta \). This means: “Cosine of 180 minus any angle is the negative of cosine of that angle”.
Example: Find cos(180°–60°)
1. Write the formula: cos(180°–θ) = -cosθ2. Substitute θ = 60°: cos(120°) = -cos(60°)
3. Value of cos(60°) is 0.5: So, cos(120°) = –0.5
Comparison: cos 0 vs cos 180
| Angle | Cos Value | Sign | Position on Unit Circle |
|---|---|---|---|
| cos 0° | 1 | Positive | Rightmost (x = +1) |
| cos 180° | -1 | Negative | Leftmost (x = -1) |
Remember: One is right, one is left. Cos 0 is +1; cos 180 is –1.
Step-by-Step Illustration: Deriving cos 180
1. Start with the unit circle (radius = 1).2. The 180-degree angle points to the leftmost edge of the circle.
3. The x-coordinate at this point is -1.
4. So, cos 180° = -1
Speed Trick or Exam Shortcut
Here's a quick trick: If the angle is exactly 180°, cos 180 is always -1. No calculation needed. For angles like (180 ± θ), instantly use –cosθ. Students use this pattern for last-minute MCQ revision, especially in chapters on trigonometry identities. Vedantu’s math sessions often start with rapid-fire quizzes on such standard trigonometric values.
Try These Yourself
- What is the value of cos 180° in radians?
- Solve: cos(180°–45°)
- Write the value of cos(180°) + sin(180°)
- Compare cos 180° and cos 0° in words.
Frequent Errors and Misunderstandings
- Confusing the sign: Many students write cos 180 = +1 instead of –1.
- Mixing with sin 180° (which is 0, not –1).
- Forgetting that cos (180 – θ) is negative.
Relation to Other Concepts
The value of cos 180 connects with topics like sine, cosine, and tangent, trigonometry tables, and the unit circle concept. Mastering this enables students to solve more advanced topics in Maths and Physics involving angles and periodicity.
Classroom Tip
A quick memory rule: "Cos zero is right (+1); cos 180 is left (–1)". Imagine the unit circle like a clock — 0° is at 3 o'clock, 180° is at 9 o'clock. Vedantu teachers often use this visual during online classes for faster sign recall.
We explored Value of cos 180 — from definition, formula, shortcut, examples, MCQs, mistakes, and links to more resources. Continue practicing with Vedantu to become confident in trigonometric questions using this easily-memorable value. For more, check the trigonometric values table or cos 360 degrees for full-circle learning.
Looking for more? Try:
- Cos 0 – direct comparison, learn positive sign story
- Trigonometric Ratios of Standard Angles – master all common sine and cosine values
- Unit Circle – see how positions on the circle define all trig values
- Cos 90 Degrees – check what happens at perpendicular
FAQs on What is the Value of Cos 180°?
1. What is the value of cos 180°?
The value of cos 180° is -1. This is a fundamental trigonometric value frequently used in various mathematical and scientific applications.
2. Why is cos 180° negative?
In the unit circle, 180° lies on the negative x-axis. The x-coordinate of any point on the unit circle represents the cosine value. Since the x-coordinate is negative in the second quadrant (where 180° lies), cos 180° is negative.
3. How is the value of cos 180° derived?
The value can be derived using several methods, including:
• Using the unit circle and observing the x-coordinate at 180°.
• Applying the cosine formula: cos(180° - θ) = -cos(θ). Substituting θ = 0°, which gives cos(180°) = -cos(0°) = -1.
• Using the trigonometric identities and the values of sine and cosine at related angles (e.g., cos(270° - 90°) ).
4. What is the difference between cos 0° and cos 180°?
Cos 0° = 1 and cos 180° = -1. They differ in sign because 0° lies on the positive x-axis while 180° lies on the negative x-axis of the unit circle.
5. What is the value of cos(180° - θ)?
The value of cos(180° - θ) = -cos θ. This is a crucial trigonometric identity for simplifying and solving trigonometric equations.
6. How do I remember the value of cos 180° quickly?
Visualize the unit circle; remember that cosine is negative in the second and third quadrants. The key angle 180° is directly opposite 0°, thus it's -1.
7. Is cos 180° the same as cos π radians?
Yes, cos 180° and cos π radians are the same, both equal to -1 because 180 degrees is equivalent to π radians.
8. What are some common mistakes students make with cos 180°?
Common errors include confusing the sign (getting +1 instead of -1) and incorrectly applying trigonometric identities involving cos 180°. Careful visualization using the unit circle helps avoid these mistakes.
9. How is cos 180° used in solving trigonometric problems?
Cos 180° = -1 is used in simplifying trigonometric expressions, solving equations, and proving trigonometric identities. It's particularly relevant in problems involving angles greater than 90°.
10. Can cos 180° be expressed as a fraction?
Yes, cos 180° = -1/1. While it's simpler to express it as -1, understanding this fractional representation can help in various algebraic manipulations.
11. What is the graphical representation of cos 180°?
On the graph of y = cos x, at x = 180°, y = -1. This point lies on the negative y-axis indicating the value of cos 180°.
12. How does understanding cos 180° help in solving real-world problems?
Understanding the value is crucial for solving problems in various fields, such as physics (wave mechanics), engineering (structural calculations), and computer graphics (creating animations). The knowledge facilitates accurate calculations of angles and distances in these contexts.
