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]
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 degrees | 0 | 1 |
| 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 degrees | 1 | 0 |
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
Therefore, for θ = 60 degrees, sin(θ/2) is ±0.5, and cos(θ/2) is ±0.866.
Most Common FAQs
Half-Angle Identities are trigonometric identities that express the sine and cosine of an angle as functions of half the angle.
These identities are commonly used to simplify complex trigonometric expressions, especially when dealing with angles that are difficult to compute directly.
Half-Angle Identities provide accurate results when the initial angle and its functions are known precisely.