Double Angle Formula Calculator

1. What is the Double Angle Formula Calculator?

Definition: This calculator computes the double angle identities for sine (\( \sin(2x) \)), cosine (\( \cos(2x) \)), and tangent (\( \tan(2x) \)) of a given angle \( x \), using the identities \( \sin(2x) = 2\sin(x)\cos(x) \), \( \cos(2x) = \cos^2(x) - \sin^2(x) \), and \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \).

Purpose: It is used in trigonometry to simplify expressions, solve equations, and find trigonometric values of double angles, often in mathematics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses the following double angle identities:

\( \sin(2x) = 2\sin(x)\cos(x) \)
\( \cos(2x) = \cos^2(x) - \sin^2(x) \), which can also be written as \( 2\cos^2(x) - 1 \) or \( 1 - 2\sin^2(x) \)
\( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \)

Where:

\( x \): The input angle in various units

Unit Conversions (Input Angle):

Angle (\( x \)):
- Degrees (deg): Directly input in degrees
- Radians (rad): Directly input in radians
- Gradians (gon): \( \text{rad} = \text{gon} \times \frac{\pi}{200} \)
- Turns (tr): \( \text{rad} = \text{tr} \times 2\pi \)
- Minutes of Arc (arcmin): \( \text{rad} = \text{deg2rad}(\text{arcmin} / 60) \)
- Seconds of Arc (arcsec): \( \text{rad} = \text{deg2rad}(\text{arcsec} / 3600) \)
- Milliradians (mrad): \( \text{rad} = \text{mrad} / 1000 \)
- Microradians (urad): \( \text{rad} = \text{urad} / 1000000 \)
- π Radians (x π rad): \( \text{rad} = \text{x π rad} \times \pi \)

Steps:

Enter the angle \( x \) and select its unit (e.g., deg, rad).
Click "Calculate" to compute the double angle values using the identities above.
The results for \( \sin(2x) \), \( \cos(2x) \), and \( \tan(2x) \) are displayed with 4 decimal places.

3. Importance of Double Angle Formulas

Double angle formulas are crucial for:

Trigonometry: Simplifying expressions and solving trigonometric equations.
Physics: Modeling periodic phenomena like waves and oscillations.
Engineering: Analyzing signals, rotations, and structural mechanics.

4. Using the Calculator

Example: Calculate the double angle identities for \( x = 30^\circ \).

Enter \( x = 30 \) and select the unit as "deg".
Click "Calculate" to compute:
- \( \sin(2x) = \sin(60^\circ) = 2\sin(30^\circ)\cos(30^\circ) = 2 \times 0.5 \times \frac{\sqrt{3}}{2} \approx 0.8660 \)
- \( \cos(2x) = \cos(60^\circ) = \cos^2(30^\circ) - \sin^2(30^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 - (0.5)^2 = 0.7500 - 0.2500 = 0.5000 \)
- \( \tan(2x) = \tan(60^\circ) = \frac{2\tan(30^\circ)}{1 - \tan^2(30^\circ)} = \frac{2 \times \frac{\sqrt{3}}{3}}{1 - \left(\frac{\sqrt{3}}{3}\right)^2} \approx 1.7321 \)

5. Frequently Asked Questions (FAQ)

Q: What are double angle formulas?
A: Double angle formulas express trigonometric functions of \( 2x \) in terms of functions of \( x \). For example, \( \sin(2x) = 2\sin(x)\cos(x) \), \( \cos(2x) = \cos^2(x) - \sin^2(x) \), and \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \).

Q: When is \( \tan(2x) \) undefined?
A: The tangent of the double angle is undefined when \( \cos(2x) = 0 \), which occurs at angles like \( 45^\circ \), \( 135^\circ \), etc., after doubling (e.g., \( x = 22.5^\circ \), \( 67.5^\circ \)).

Q: What are the different angle units?
A: Angles can be measured in various units:

Degrees (deg): 360° in a full circle.
Radians (rad): \( 2\pi \) in a full circle.
Gradians (gon): 400 gon in a full circle.
Turns (tr): 1 turn is a full circle.
Minutes of Arc (arcmin): 60 arcmin per degree.
Seconds of Arc (arcsec): 3600 arcsec per degree.
Milliradians (mrad): 1000 mrad per radian.
Microradians (urad): 1000000 urad per radian.
π Radians (x π rad): Expressed as a multiple of π.