1. What is the Sum and Difference Identities Calculator?

Definition: This calculator computes the sine, cosine, tangent, cotangent, secant, and cosecant of the sum (\( \alpha + \beta \)) and difference (\( \alpha - \beta \)) of two angles, using the following identities:

• \( \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \)
• \( \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \)
• \( \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \)
• \( \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \)
• \( \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)} \)
• \( \tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)} \)
• \( \cot(\alpha + \beta) = \frac{\cot(\alpha)\cot(\beta) - 1}{\cot(\beta) + \cot(\alpha)} \)
• \( \cot(\alpha - \beta) = \frac{\cot(\alpha)\cot(\beta) + 1}{\cot(\beta) - \cot(\alpha)} \)
• \( \sec(\alpha + \beta) = \frac{\sec(\alpha)\sec(\beta)\csc(\alpha)\csc(\beta)}{\csc(\alpha)\csc(\beta) - \sec(\alpha)\sec(\beta)} \)
• \( \sec(\alpha - \beta) = \frac{\sec(\alpha)\sec(\beta)\csc(\alpha)\csc(\beta)}{\csc(\alpha)\csc(\beta) + \sec(\alpha)\sec(\beta)} \)
• \( \csc(\alpha + \beta) = \frac{\sec(\alpha)\sec(\beta)\csc(\alpha)\csc(\beta)}{\sec(\alpha)\csc(\beta) + \csc(\alpha)\sec(\beta)} \)
• \( \csc(\alpha - \beta) = \frac{\sec(\alpha)\sec(\beta)\csc(\alpha)\csc(\beta)}{\sec(\alpha)\csc(\beta) - \csc(\alpha)\sec(\beta)} \)

Purpose: These identities simplify trigonometric expressions and solve problems in mathematics, physics, engineering, and other fields where angles are combined or subtracted, such as in wave interference or rotations.

2. How Does the Calculator Work?

The calculator uses the sum and difference identities to compute the trigonometric functions of \( \alpha + \beta \) and \( \alpha - \beta \).

Steps:

• Enter the angles \( \alpha \) and \( \beta \), and select their units (degrees, radians, etc.).
• The calculator converts both angles to radians, computes the necessary trigonometric functions for \( \alpha \) and \( \beta \), and applies the identities to find the results.
• Results are displayed with 5 decimal places.

Unit Conversions (Angles):

• Degrees: \( \text{rad} = \text{deg} \times \frac{\pi}{180} \)
• Radians: No conversion needed
• Gradians: \( \text{rad} = \text{gon} \times \frac{\pi}{200} \)
• Turns: \( \text{rad} = \text{tr} \times 2\pi \)
• Minutes of Arc: \( \text{rad} = \text{deg2rad}(\text{arcmin} / 60) \)
• Seconds of Arc: \( \text{rad} = \text{deg2rad}(\text{arcsec} / 3600) \)
• Milliradians: \( \text{rad} = \text{mrad} / 1000 \)
• Microradians: \( \text{rad} = \text{urad} / 1000000 \)
• π Radians: \( \text{rad} = \text{x π rad} \times \pi \)

3. Properties of the Sum and Difference Identities

These identities are fundamental in trigonometry and are derived from the angle addition formulas on the unit circle. They can be used to simplify expressions, solve equations, or compute trigonometric values of combined angles.

4. Using the Calculator

Example 1: Calculate the trigonometric functions for \( \alpha = 45^\circ \) and \( \beta = 30^\circ \).

1. Enter \( \alpha = 45 \), with unit "degrees".
2. Enter \( \beta = 30 \), with unit "degrees".
3. Click "Calculate" to compute:
   • \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.70711 \), \( \cos(45^\circ) \approx 0.70711 \)
   • \( \sin(30^\circ) = 0.5 \), \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.86603 \)
   • \( \tan(45^\circ) = 1 \), \( \tan(30^\circ) = \frac{\sqrt{3}}{3} \approx 0.57735 \)
   • \( \cot(45^\circ) = 1 \), \( \cot(30^\circ) = \sqrt{3} \approx 1.73205 \)
   • \( \sec(45^\circ) = \sqrt{2} \approx 1.41421 \), \( \sec(30^\circ) = \frac{2}{\sqrt{3}} \approx 1.15470 \)
   • \( \csc(45^\circ) = \sqrt{2} \approx 1.41421 \), \( \csc(30^\circ) = 2 \)
   • \( \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) \approx 0.96593 \)
   • \( \sin(45^\circ - 30^\circ) = \sin(45^\circ)\cos(30^\circ) - \cos(45^\circ)\sin(30^\circ) \approx 0.25882 \)
   • \( \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ) \approx 0.25882 \)
   • \( \cos(45^\circ - 30^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ) \approx 0.96593 \)
   • \( \tan(45^\circ + 30^\circ) = \frac{\tan(45^\circ) + \tan(30^\circ)}{1 - \tan(45^\circ)\tan(30^\circ)} \approx 3.73205 \)
   • \( \tan(45^\circ - 30^\circ) = \frac{\tan(45^\circ) - \tan(30^\circ)}{1 + \tan(45^\circ)\tan(30^\circ)} \approx 0.26795 \)
   • \( \cot(45^\circ + 30^\circ) = \frac{\cot(45^\circ)\cot(30^\circ) - 1}{\cot(30^\circ) + \cot(45^\circ)} \approx 0.26795 \)
   • \( \cot(45^\circ - 30^\circ) = \frac{\cot(45^\circ)\cot(30^\circ) + 1}{\cot(30^\circ) - \cot(45^\circ)} \approx 3.73205 \)
   • \( \sec(45^\circ + 30^\circ) = \frac{\sec(45^\circ)\sec(30^\circ)\csc(45^\circ)\csc(30^\circ)}{\csc(45^\circ)\csc(30^\circ) - \sec(45^\circ)\sec(30^\circ)} \approx 3.86370 \)
   • \( \sec(45^\circ - 30^\circ) = \frac{\sec(45^\circ)\sec(30^\circ)\csc(45^\circ)\csc(30^\circ)}{\csc(45^\circ)\csc(30^\circ) + \sec(45^\circ)\sec(30^\circ)} \approx 1.03528 \)
   • \( \csc(45^\circ + 30^\circ) = \frac{\sec(45^\circ)\sec(30^\circ)\csc(45^\circ)\csc(30^\circ)}{\sec(45^\circ)\csc(30^\circ) + \csc(45^\circ)\sec(30^\circ)} \approx 1.03528 \)
   • \( \csc(45^\circ - 30^\circ) = \frac{\sec(45^\circ)\sec(30^\circ)\csc(45^\circ)\csc(30^\circ)}{\sec(45^\circ)\csc(30^\circ) - \csc(45^\circ)\sec(30^\circ)} \approx 3.86370 \)

Example 2: Calculate the trigonometric functions for \( \alpha = \frac{\pi}{4} \) and \( \beta = \frac{\pi}{6} \), with output in radians.

1. Enter \( \alpha = \frac{\pi}{4} \), with unit "radians".
2. Enter \( \beta = \frac{\pi}{6} \), with unit "radians".
3. Click "Calculate" to compute (results are the same as Example 1, since \( \frac{\pi}{4} = 45^\circ \) and \( \frac{\pi}{6} = 30^\circ \)).

5. Frequently Asked Questions (FAQ)

Q: What are the sum and difference identities?
A: These are trigonometric formulas that express the sine, cosine, tangent, cotangent, secant, and cosecant of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. They are fundamental tools in trigonometry.

Q: When are these functions undefined?
A: Functions like \( \tan(\alpha + \beta) \) are undefined when the denominator is zero (e.g., \( 1 - \tan(\alpha)\tan(\beta) = 0 \)). Similarly, \( \sec \) and \( \csc \) are undefined when \( \cos \) or \( \sin \) is zero, respectively.

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

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

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