How to Calculate Sin 120 Degrees Using the Unit Circle?
The concept of Sin 120 Degrees plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding sin 120 is essential for solving trigonometry questions, geometry problems, and even for entrance exams like JEE and NEET. This value frequently appears in triangles, physics, and engineering tasks, making it a must-know for students.
What Is Sin 120 Degrees?
Sin 120 Degrees is the value of the sine trigonometric function for an angle equal to 120°. It describes the ratio of the opposite side to hypotenuse in a right triangle, and it also represents the y-coordinate of a point on the unit circle at 120°. You’ll find this concept applied in areas such as coordinate geometry, trigonometric ratios, and unit circle analysis.
Key Formula for Sin 120 Degrees
Here’s the standard formula: \( \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \)
Value of Sin 120 Degrees (Quick Reference Table)
| Angle (Degrees) | Sin Value (Fraction) | Decimal Approx. |
|---|---|---|
| 60° | √3/2 | 0.866 |
| 120° | √3/2 | 0.866 |
| 180° | 0 | 0 |
Cross-Disciplinary Usage
Sin 120 degrees is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. For example, it helps with signal processing in engineering, analyzing projectile motion in physics, and even programming graphics. Students preparing for JEE or NEET will see its relevance in various trigonometry and geometry questions.
Step-by-Step Derivation: How to Find Sin 120°
Let’s derive the value of sin 120° using identities and the unit circle:
- Recognize the related acute angle: 120° = 180° − 60°
So, use the identity: \( \sin(180^\circ - \theta) = \sin \theta \) - Apply the formula:
\( \sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin 60^\circ \) - Recall \( \sin 60^\circ \):
\( \sin 60^\circ = \frac{\sqrt{3}}{2} \) - Final Answer:
\( \sin 120^\circ = \frac{\sqrt{3}}{2} \)
Sin 120° via the Unit Circle
On the unit circle, 120° is in the second quadrant where sine values are positive. The coordinates corresponding to 120° are \((-1/2, \sqrt{3}/2)\). Therefore, sin 120° is exactly \( \sqrt{3}/2 \), which is the y-coordinate at that angle.
Example Problems With Solutions
1. Find the value of sin 120° - cos 30°
1. sin 120° = \( \frac{\sqrt{3}}{2} \)2. cos 30° = \( \frac{\sqrt{3}}{2} \)
3. sin 120° - cos 30° = \( \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = 0 \)
2. Evaluate 4 × sin 120° + tan 45°
1. sin 120° = \( \frac{\sqrt{3}}{2} \)2. tan 45° = 1
3. 4 × \( \frac{\sqrt{3}}{2} \) = \( 2\sqrt{3} \)
4. \( 2\sqrt{3} + 1 \) is the answer.
Speed Trick or Vedic Shortcut
A quick way to remember sin 120° is to use the “Supplement Rule” — any angle of the form \( \sin(180^\circ - x) \) has the same sine value as \( \sin x \). Since sin 120° = sin 60°, you just need to recall sin 60°, which is \( \sqrt{3}/2 \).
Tip: Remember: All sine values in the second quadrant (between 90° and 180°) are positive!
Try These Yourself
- Calculate sin 120° using the unit circle definition.
- Show that sin 120° = cos 30°.
- Find all angles between 0° and 360° with the same sine value as sin 120°.
- Solve: 2 × sin 120° + cos 60° = ?
Frequent Errors and Misunderstandings
- Confusing sin 120° with cos 120° (cos 120° is negative).
- Forgetting that sine is positive in the second quadrant.
- Using calculator in radians instead of degrees (double check mode!).
Relation to Other Concepts
The idea of Sin 120 Degrees connects closely with topics such as Sin 60 Degrees and the Unit Circle. Mastering this helps with understanding trigonometric identities like \( \sin(180^\circ - x) = \sin x \), symmetry on the unit circle, and solving triangle area problems.
Classroom Tip
A fast way to remember sin 120° is the “mirror” memory: 120° is 60° away from 180°, so sine remains the same as 60°. Vedantu’s teachers often use quadrant charts and color-coding to help you visualise the sign and value of sine for different angles during live classes.
We explored Sin 120 Degrees—from definition, formula, stepwise derivation, examples, common mistakes, and its connection to other maths concepts. Continue practicing with Vedantu and review related resources like the Sin Cos Tan Table or more Trigonometry Angle Values to strengthen your trigonometric skills.
For deeper understanding and practice questions featuring sin 120°, check out Trigonometric Identities for exam-oriented revision.
FAQs on What is the Value and Formula of Sin 120 Degrees?
1. What is the value of sin 120 degrees?
The value of sin 120 degrees is √3/2, or approximately 0.866. This is because sin 120° is equivalent to sin (180° - 60°) which simplifies to sin 60°, and sin 60° = √3/2.
2. How do I find sin 120 degrees using the unit circle?
On the unit circle, 120 degrees lies in the second quadrant. The sine value corresponds to the y-coordinate of the point on the circle at that angle. This y-coordinate is √3/2.
3. What is the relationship between sin 120 degrees and other trigonometric functions?
Sin 120° can be expressed using several identities. For example: sin 120° = sin (180° - 60°) = cos 30° = √3/2. You can also use the identity sin θ = cos (90° - θ) or other relevant trigonometric identities to derive it.
4. How can I remember the value of sin 120 degrees quickly?
Remember that sin 120° = sin (180° - 60°) = sin 60° = √3/2. Focus on the relationship with the well-known value of sin 60°. Visualizing the unit circle can also be helpful.
5. Is sin 120 degrees positive or negative?
Sin 120° is positive because 120° lies in the second quadrant, where the sine function is positive.
6. How do I calculate sin 120 degrees without a calculator?
Use the identity sin 120° = sin (180° - 60°) = sin 60° = √3/2. You should know the values of trigonometric functions for standard angles like 30°, 45°, and 60°.
7. What is sin 120 degrees in terms of radians?
120 degrees is equal to (2π/3) radians. Therefore, sin 120° = sin (2π/3) = √3/2.
8. Where do we use the value of sin 120 degrees?
The value of sin 120° is used in various applications involving triangles, such as solving trigonometric equations, calculating lengths and angles in triangles, and in various mathematical and engineering problems.
9. What is the difference between sin 60 degrees and sin 120 degrees?
Numerically, sin 60° and sin 120° are equal (√3/2). However, they represent different angles on the unit circle, lying in the first and second quadrants respectively. Their reference angles are the same.
10. Can sin 120 degrees be expressed as a fraction?
Yes, sin 120° can be expressed as the fraction √3/2.
11. How is sin 120 degrees related to the cosine function?
Using the cofunction identity, sin 120° = cos (90° - 120°) = cos (-30°) = cos 30° = √3/2.
12. What are some common mistakes students make when calculating sin 120 degrees?
A common mistake is forgetting that sine is positive in the second quadrant, leading to an incorrect negative value. Another mistake is incorrectly applying trigonometric identities or failing to simplify the result correctly.
