Sin 120 Degrees Value, Calculation, and Applications - RRTutors

What is the Value of Sin 120 Degrees?

The sine of 120 degrees equals √3/2 (approximately 0.866).

When we look at the unit circle, sin 120° corresponds to the y-coordinate of the point where the angle of 120° intersects the circle. This places us in the second quadrant, where sine values are positive.

Understanding Sin 120° Through Calculation

We can find sin 120° in several ways:

Using the Unit Circle

In the unit circle, sin 120° equals the y-coordinate of the point (x,y) where a 120° angle intersects the circle. Since 120° is in the second quadrant, we can use reference angles.

120° = 180° - 60°, so sin 120° = sin(180° - 60°) = sin 60° = √3/2

Using Trigonometric Identities

• sin(180° - θ) = sin θ
• sin 120° = sin(180° - 60°) = sin 60° = √3/2

Special Properties of Sin 120°

Sin 120° is one of the standard angles in trigonometry with an exact value. At 120°, we have:

• sin 120° = √3/2 ≈ 0.866
• cos 120° = -1/2
• tan 120° = -√3

Applications of Sin 120°

Sin 120° appears frequently in:

1. Physics: Force vector calculations, especially when analyzing forces at 120° angles
2. Engineering: AC circuits with three-phase systems (phases separated by 120°)
3. Geometry: Regular hexagons have interior angles of 120°
4. Crystallography: Many crystal structures have 120° bond angles

Common Questions About Sin 120°

Is Sin 120° Positive or Negative?

Sin 120° is positive because 120° falls in the second quadrant of the unit circle, where sine values are positive.

How to Convert Sin 120° to Radians?

120 degrees = (120 × π/180) = 2π/3 radians Therefore, sin 120° = sin(2π/3) = √3/2

What is Sin 120° as a Decimal?

Sin 120° = √3/2 ≈ 0.866025404

Sin 120° in a Calculator

Most scientific calculators can directly calculate sin 120° by:

1. Setting the calculator to degree mode
2. Entering 120
3. Pressing the sin button

Summary

Sin 120° equals √3/2 (approximately 0.866), a positive value found in the second quadrant of the unit circle. This value appears frequently in mathematics, physics, and engineering applications