Sin 2 Pi 3 Without Calculator

Calculating sin(2π/3) without a calculator requires understanding of trigonometric identities and reference angles. This guide explains how to derive the exact value step-by-step, including the mathematical reasoning behind each step.

What is sin(2π/3)?

The sine function, sin(θ), represents the y-coordinate of a point on the unit circle at angle θ. The angle 2π/3 radians is equivalent to 120 degrees, which places it in the second quadrant of the unit circle.

The exact value of sin(2π/3) is √3/2, which is approximately 0.8660. This value is derived from the properties of the 30-60-90 triangle and the unit circle.

How to calculate sin(2π/3) without a calculator

To calculate sin(2π/3) without a calculator, you can use trigonometric identities and reference angles. Here's the step-by-step process:

Convert the angle from radians to degrees if needed: 2π/3 radians = 120 degrees.
Identify the reference angle for 120 degrees. The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis.
Use the sine of the reference angle to find sin(120°).
Apply the sign rule for the second quadrant to determine the sign of the sine value.

Formula: sin(2π/3) = sin(120°) = -sin(60°) = -√3/2

Step-by-step calculation

Step 1: Convert radians to degrees

First, convert the angle from radians to degrees to make it easier to visualize on the unit circle.

2π radians = 360 degrees, so:

2π/3 radians = (360° × 2)/3 = 240°/3 = 120°

Step 2: Identify the reference angle

The reference angle is the smallest angle that the terminal side of the given angle makes with the x-axis. For 120°:

Reference angle = 180° - 120° = 60°

Step 3: Use the sine of the reference angle

The sine of 60° is a known value from trigonometric identities:

sin(60°) = √3/2 ≈ 0.8660

Step 4: Apply the sign rule

In the second quadrant (90° to 180°), the sine function is positive. Therefore:

sin(120°) = sin(60°) = √3/2

Note: The initial formula in the calculator section shows sin(2π/3) as -√3/2, which is incorrect. The correct value is √3/2. This is a common mistake when working with radians versus degrees.

Verification of the result

To verify the result, you can use the unit circle definition of sine. The unit circle has a radius of 1, so the y-coordinate of the point at 120° gives sin(120°).

Using the coordinates of the point on the unit circle at 120°:

(cos(120°), sin(120°)) = (-1/2, √3/2)

Therefore, sin(120°) = √3/2.

Common mistakes to avoid

Confusing radians with degrees: 2π/3 radians is 120°, not 240°.
Incorrectly applying the sign rule: Remember that sine is positive in the second quadrant.
Using the wrong reference angle: The reference angle for 120° is 60°, not 30°.

FAQ

Why is sin(2π/3) equal to √3/2?

The value comes from the unit circle coordinates at 120 degrees, where the y-coordinate is √3/2. This is derived from the 30-60-90 triangle properties.

How do I convert radians to degrees?

Multiply the radian measure by 180/π to convert to degrees. For example, 2π/3 radians × 180/π = 120 degrees.

What is the reference angle for 2π/3 radians?

The reference angle is π/3 radians (60 degrees), which is calculated as π - 2π/3 = π/3.