Why Is Cos 120° Negative and How Is It Calculated?
The concept of value of cos 120 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing the value of trigonometric functions at standard angles like 120 degrees is essential for quick calculations in school, competitive exams, and science subjects.
What Is Value of Cos 120?
The value of cos 120 refers to the cosine of a 120-degree angle. In trigonometry, cosine represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cos 120 is especially important in topics like the unit circle, trigonometric ratios, and when working with angles in standard intervals. You’ll find this concept applied in areas such as geometry problems, vectors, and graphical representation of trigonometric functions.
Key Formula for Value of Cos 120
Here’s the standard formula: \( \cos(120^{\circ}) = \cos(180^{\circ} - 60^{\circ}) = -\cos(60^{\circ}) = -\frac{1}{2} \)
Cross-Disciplinary Usage
The value of cos 120 is not only useful in Maths but also plays an important role in Physics, Computer Science, and logical reasoning. When working with vectors, geometry, or even circuit analysis, cos 120 can appear in calculations. Students preparing for JEE, NEET, or board exams will see its relevance in various types of questions and word problems.
Step-by-Step Illustration
- Start with the standard angle: \( \cos(120^\circ) \)
- Express 120° as (180° – 60°):
\( \cos(120^\circ) = \cos(180^\circ - 60^\circ) \) - Use the trigonometric identity \( \cos(180^\circ-\theta) = -\cos\theta \):
\( \cos(120^\circ) = -\cos(60^\circ) \) - Recall that \( \cos(60^\circ) = \frac{1}{2} \):
So \( \cos(120^\circ) = -\frac{1}{2} \) - Final Answer: Cos 120° = -½ or -0.5
Standard Values Table for Cosine
| Angle (°) | Cosine Value |
|---|---|
| 0 | 1 |
| 30 | √3/2 |
| 45 | 1/√2 |
| 60 | 1/2 |
| 90 | 0 |
| 120 | -1/2 |
| 180 | -1 |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for remembering the value of cos 120:
- If the angle is 120°, notice that it’s 60° past 60°, i.e., 120° = 180° – 60°.
- In the second quadrant (between 90° and 180°), cosine is negative.
- So, just take the positive cosine of 60° (which is ½) and put a minus sign.
- Answer: cos 120° = -½
Tricks like quadrant sign rules (All Students Take Calculus — CAST) help you quickly determine signs for all trigonometric values. Vedantu’s live and recorded sessions offer many such fast revision strategies for board and entrance exams.
Try These Yourself
- Calculate cos 150° using a similar method as cos 120°.
- Find the value of cos 240°.
- What is cos(180° – x) in terms of cos x?
- Is the value of cos 120° equal to the value of sin 30°? Explain why.
Frequent Errors and Misunderstandings
- Thinking the value of cos 120 is positive because 120 is less than 180 (remember: in the second quadrant, only sine is positive).
- Forgetting to use a negative sign for cosine between 90° and 180°.
- Confusing cos 120° with cos 60° — always check the quadrant and sign.
Relation to Other Concepts
The idea of value of cos 120 connects closely with topics such as trigonometric tables, identities, and complimentary angles. Mastering this helps you with triangle problems, vector calculations, and transformations in advanced classes.
Classroom Tip
A quick way to remember the value of cos 120 is to always visualize the unit circle and recall that in the second quadrant, cosine values are always negative. Vedantu’s teachers often draw the CAST diagram (All Students Take Calculus) to help students assign signs correctly and quickly solve trigonometric questions.
We explored the value of cos 120—from its definition and formula to its stepwise calculation, common mistakes, and how it ties in with other trigonometric ideas. Practice more trigonometry with Vedantu’s expert resources and interactive sessions to become confident at using these values in any context!
Internal Links to Related Topics
- Trigonometric Values of Standard Angles
- Unit Circle Explanation
- Trigonometric Identities
- Sin, Cos, Tan Values Table
FAQs on What is the Value of Cos 120° in Trigonometry?
1. What is the exact value of cos 120° in trigonometry?
The exact value of cos 120° is -1/2. In decimal form, this is equal to -0.5. This value is derived from its position in the second quadrant of the unit circle, where the cosine function's value is always negative.
2. How can the value of cos 120° be calculated using trigonometric identities?
The value of cos 120° can be calculated using standard trigonometric identities as per the CBSE 2025-26 syllabus. Two common methods are:
- Using the identity cos(180° - θ) = -cos(θ): We can write cos 120° as cos(180° - 60°). This simplifies to -cos(60°), which equals -1/2.
- Using the identity cos(90° + θ) = -sin(θ): We can also express cos 120° as cos(90° + 30°). This simplifies to -sin(30°), which also equals -1/2.
3. Why is the value of cos 120° negative?
The value of cos 120° is negative because the angle 120° lies in the second quadrant (an angle between 90° and 180°). In the Cartesian coordinate system, the cosine of an angle represents the x-coordinate on a unit circle. For any angle in the second quadrant, the x-coordinate is always negative. According to the CAST rule, only Sine is positive in the second quadrant.
4. Is the value of cos 120° the same as sin 30°? Explain why or why not.
No, the value of cos 120° is not the same as sin 30°, which is a common point of confusion. Although their absolute values are equal, their signs are different.
- The value of cos 120° is -1/2 because it falls in the second quadrant where cosine is negative.
- The value of sin 30° is +1/2 because it falls in the first quadrant where all trigonometric functions are positive.
5. What is a simple trick to remember the sign of cos 120°?
A simple and effective trick is the "All Students Take Calculus" (ASTC) mnemonic, which indicates which functions are positive in each quadrant.
- A (Quadrant I): All functions are positive.
- S (Quadrant II): Sine (and its reciprocal, cosecant) are positive.
- T (Quadrant III): Tangent (and its reciprocal, cotangent) are positive.
- C (Quadrant IV): Cosine (and its reciprocal, secant) are positive.
6. In which subjects besides mathematics, like Physics, is the value of cos 120° important?
The value of cos 120° is crucial in many areas of Physics. For instance, it is used when calculating the work done by a force when the force and displacement vectors are at an angle of 120° to each other (using the formula W = Fd cos θ). It is also fundamental in problems involving vector addition, like finding the resultant of two forces, and in analysing alternating current (AC) circuits with phase differences.
7. What is the most common mistake students make when finding the value of cos 120° in exams?
The most common mistake is a sign error. Students often correctly determine the reference angle (60°) and its cosine value (1/2) but forget to apply the negative sign appropriate for the second quadrant. Another frequent error is misremembering the allied angle formulas, for example, incorrectly equating cos(180° - 60°) with +cos(60°). Always confirming the angle's quadrant is the best way to avoid this mistake.
