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Sin 2θ Calculator

Sine of a Double Angle

The formula for the sine of a double angle, denoted as sin(2θ), allows us to determine the sine of an angle that is twice a given angle θ. This trigonometric identity is especially useful for simplifying expressions and solving problems involving double angles. The formula is given by:

sin(2θ) = 2sin(θ)cos(θ)

How to Calculate sin(2θ):

  1. Identify θ: Start with the angle θ for which you need to find the sine of the double angle.
  2. Find sin(θ): Determine the sine of the angle θ. This can be done using a calculator, trigonometric tables, or known values from special angles (like 30°, 45°, 60°, etc.).
  3. Find cos(θ): Determine the cosine of the angle θ. Similar to finding the sine, this can be achieved through various methods including a calculator or reference to known values.
  4. Multiply and Double: Once you have sin(θ) and cos(θ), multiply these two values together and then double the result. Mathematically, this step is expressed as:

sin(2θ) = 2 × sin(θ) × cos(θ)

Example Calculation:

Let’s say we want to calculate sin(2θ) for θ = 30°:

  1. Determine sin(30°):
    sin(30°) = 1/2
  2. Determine cos(30°):
    cos(30°) = √3/2
  3. Apply the double angle formula:
    sin(2 × 30°) = sin(60°)
    sin(2 × 30°) = 2 × sin(30°) × cos(30°)
    sin(60°) = 2 × 1/2 × √3/2
    sin(60°) = √3/2

So, sin(60°) = √3/2, confirming our calculation is consistent with known values.

In summary, the formula sin(2θ) = 2sin(θ)cos(θ) provides a straightforward way to compute the sine of a double angle using the sine and cosine of the original angle θ.

Alternative Methods to Calculate sin(2θ)

Using the Pythagorean Identity

Given the identities sin²(θ) + cos²(θ) = 1 and the double angle identity for cosine, cos(2θ) = cos²(θ) - sin²(θ), we can derive another form of sin(2θ):

  1. From cos(2θ) = 1 - 2sin²(θ) and cos(2θ) = 2cos²(θ) - 1:

    cos(2θ) = 1 - 2sin²(θ)

    cos(2θ) = 2cos²(θ) - 1

  2. Since sin(2θ) and cos(2θ) are related through the Pythagorean identity sin²(2θ) + cos²(2θ) = 1, we can express sin(2θ) in terms of cos(2θ):

    sin(2θ) = ±√(1 - cos²(2θ))

  3. Using cos(2θ) = 1 - 2sin²(θ):

    sin(2θ) = ±√(1 - (1 - 2sin²(θ))²)

Using Euler’s Formula

Euler’s formula relates complex exponentials to trigonometric functions: e^(iθ) = cos(θ) + i sin(θ).

  1. Start with Euler’s formula for a single angle:

    e^(iθ) = cos(θ) + i sin(θ)

  2. Square both sides to get the double angle:

    (e^(iθ))² = (cos(θ) + i sin(θ))²

    e^(i2θ) = cos²(θ) - sin²(θ) + 2i sin(θ) cos(θ)

  3. Equate the imaginary parts on both sides:

    2i sin(θ) cos(θ) = i sin(2θ)

    Therefore, sin(2θ) = 2 sin(θ) cos(θ)

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