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Sin 120 Degrees Without Calculator

Reviewed by Calculator Editorial Team

Calculating sin 120 degrees without a calculator requires understanding of trigonometric identities and properties of the unit circle. This guide explains three reliable methods to find the sine of 120° using fundamental trigonometric principles.

How to Calculate sin 120° Without a Calculator

There are several methods to find sin 120° without a calculator. The most common approaches include using trigonometric identities, the unit circle method, and reference angles. Each method provides the same result but uses different conceptual frameworks.

Key Formula

The sine of an angle in the second quadrant can be found using the identity:

sin(180° - θ) = sinθ

For 120°, this becomes sin(180° - 60°) = sin60°

This identity shows that sin 120° is equal to sin 60°, which is √3/2 or approximately 0.8660.

Using Trigonometric Identities

The first method uses the sine of supplementary angles identity. Since 120° and 60° are supplementary angles (they add up to 180°), their sines are equal.

Important Note

This identity only works for angles in the range of 0° to 180°. For angles outside this range, you would need to consider the periodicity of the sine function.

Steps to calculate sin 120° using this method:

  1. Recognize that 120° = 180° - 60°
  2. Apply the identity sin(180° - θ) = sinθ
  3. Calculate sin 60° which is √3/2
  4. Therefore, sin 120° = sin 60° = √3/2

Unit Circle Method

The unit circle method involves plotting the angle on the unit circle and determining the corresponding y-coordinate, which represents the sine value.

Steps to calculate sin 120° using the unit circle:

  1. Draw the unit circle with radius 1
  2. Mark the angle of 120° from the positive x-axis
  3. The y-coordinate of the point where the terminal side intersects the circle is sin 120°
  4. Since 120° is in the second quadrant, the y-coordinate is positive
  5. The reference angle is 60° (180° - 120°)
  6. The y-coordinate for 60° is √3/2
  7. Therefore, sin 120° = √3/2

Reference Angle Approach

The reference angle method involves finding the reference angle and using the appropriate trigonometric sign based on the quadrant.

Steps to calculate sin 120° using the reference angle:

  1. Determine the quadrant: 120° is in the second quadrant (90° < 120° < 180°)
  2. Find the reference angle: 180° - 120° = 60°
  3. In the second quadrant, sine is positive
  4. Calculate sin 60° which is √3/2
  5. Therefore, sin 120° = sin 60° = √3/2

Example Calculation

Let's work through an example to find sin 120° using the trigonometric identity method.

Worked Example

Find sin 120° using the identity sin(180° - θ) = sinθ.

  1. Let θ = 60° because 120° = 180° - 60°
  2. Apply the identity: sin 120° = sin(180° - 60°) = sin 60°
  3. We know that sin 60° = √3/2 ≈ 0.8660
  4. Therefore, sin 120° = √3/2 ≈ 0.8660

This example demonstrates how to apply the trigonometric identity to find the sine of 120 degrees.

Frequently Asked Questions

Why is sin 120° equal to sin 60°?

Because 120° and 60° are supplementary angles (they add up to 180°), their sines are equal according to the trigonometric identity sin(180° - θ) = sinθ.

Can I use the unit circle method for any angle?

Yes, the unit circle method can be used for any angle. You simply plot the angle on the unit circle and read the corresponding y-coordinate for the sine value.

What is the reference angle for 120°?

The reference angle for 120° is 60° because it's the acute angle formed between the terminal side of 120° and the x-axis.

Is sin 120° positive or negative?

Sin 120° is positive because 120° is in the second quadrant where sine values are positive.

How accurate are these methods?

These methods are mathematically precise and will always give the exact value of sin 120° as √3/2 or approximately 0.8660.