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Half Angle Identities Calculator Online

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The Half Angle Identities Calculator is a tool that simplifies trigonometric calculations by determining the sine and cosine of half angles, expressed as functions of the original angle. It’s particularly useful in trigonometry, aiding in simplifying complex expressions involving trigonometric functions.

Formula of Half Angle Identities Calculator

The calculator uses a straightforward formula which is as follows:

Sine Half-Angle Identity:

The formula for the sine half-angle identity is:

sin(θ/2) = ±√[(1 – cos(θ)) / 2]

Cosine Half-Angle Identity:

The formula for the cosine half-angle identity is:

cos(θ/2) = ±√[(1 + cos(θ)) / 2]

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General Terms Table

Here’s a quick reference table for general terms often searched, providing a helpful reference without the need for manual calculation:

Original Angle (θ)Sine Half-Angle (sin(θ/2))Cosine Half-Angle (cos(θ/2))
0 degrees01
30 degrees±√(3 – √3) / 2√((2 + √3) / 4)
45 degrees±√((1 – √2) / 2)√((2 + √2) / 4)
60 degrees±√(3 + √3) / 2√((2 – √3) / 4)
90 degrees10

Example of Half Angle Identities Calculator

Let’s take an example to understand the application of the half-angle identities. Consider an angle θ = 60 degrees.

Using the formulas:

  • sin(60/2) = ±√[(1 – cos(60)) / 2] = ±√[(1 – 0.5) / 2] = ±√[0.25] = ±0.5
  • cos(60/2) = ±√[(1 + cos(60)) / 2] = ±√[(1 + 0.5) / 2] = ±√[0.75] = ±0.866
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Therefore, for θ = 60 degrees, sin(θ/2) is ±0.5, and cos(θ/2) is ±0.866.

Most Common FAQs

1. What are Half-Angle Identities?

Half-Angle Identities are trigonometric identities that express the sine and cosine of an angle as functions of half the angle.

2. When are Half-Angle Identities Used?

These identities are commonly used to simplify complex trigonometric expressions, especially when dealing with angles that are difficult to compute directly.

3. How Accurate are Half-Angle Identity Calculations?

Half-Angle Identities provide accurate results when the initial angle and its functions are known precisely.

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