Angle Addition Identities Calculator
Calculate trigonometric values using angle addition or subtraction identities: sin(A±B), cos(A±B), tan(A±B).
What is an Angle Addition Identities Calculator?
An angle addition identities calculator is a tool used to find the trigonometric value (sine, cosine, or tangent) of the sum or difference of two angles (A and B), using the values of the trigonometric functions of the individual angles A and B. These identities are fundamental in trigonometry and are used to simplify expressions, solve equations, and find exact trigonometric values for angles that can be expressed as the sum or difference of standard angles (like 30°, 45°, 60°, 90°).
Anyone studying trigonometry, calculus, physics, or engineering will find the angle addition identities calculator useful. It helps in understanding how trigonometric functions behave when angles are combined.
Common misconceptions include thinking these identities apply to the functions themselves being added (e.g., sin(A+B) is NOT sin(A) + sin(B)). The angle addition identities calculator correctly applies the specific formulas.
Angle Addition Identities Formula and Mathematical Explanation
The angle addition and subtraction identities are as follows:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- sin(A – B) = sin(A)cos(B) – cos(A)sin(B)
- cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
- cos(A – B) = cos(A)cos(B) + sin(A)sin(B)
- tan(A + B) = (tan(A) + tan(B)) / (1 – tan(A)tan(B))
- tan(A – B) = (tan(A) – tan(B)) / (1 + tan(A)tan(B))
These identities can be derived geometrically using the unit circle or vector rotations. For example, the cosine of the difference of two angles can be found using the distance formula between two points on the unit circle corresponding to angles A and B.
The angle addition identities calculator uses these exact formulas based on your selection.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| A | The first angle | Degrees (or Radians) | Any real number |
| B | The second angle | Degrees (or Radians) | Any real number |
| sin(A), cos(A), tan(A) | Trig functions of angle A | Dimensionless | sin, cos: [-1, 1], tan: (-∞, ∞) |
| sin(B), cos(B), tan(B) | Trig functions of angle B | Dimensionless | sin, cos: [-1, 1], tan: (-∞, ∞) |
| sin(A±B), cos(A±B), tan(A±B) | Trig functions of the sum/difference | Dimensionless | sin, cos: [-1, 1], tan: (-∞, ∞) |
Table of variables used in the angle addition identities calculator.
Practical Examples (Real-World Use Cases)
Example 1: Finding sin(75°)
We want to find sin(75°). We can express 75° as 45° + 30°. Using the angle addition identities calculator (or formula): sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°) We know sin(45°) = √2/2, cos(30°) = √3/2, cos(45°) = √2/2, sin(30°) = 1/2. So, sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4 ≈ (2.449 + 1.414) / 4 ≈ 0.9659.
Example 2: Finding cos(15°)
We want to find cos(15°). We can express 15° as 45° – 30° (or 60° – 45°). Using the angle addition identities calculator (or formula for cos(A-B)): cos(15°) = cos(45° – 30°) = cos(45°)cos(30°) + sin(45°)sin(30°) cos(15°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4 ≈ 0.9659.
How to Use This Angle Addition Identities Calculator
- Enter Angle A: Input the value for the first angle (A) in degrees into the "Angle A" field.
- Enter Angle B: Input the value for the second angle (B) in degrees into the "Angle B" field.
- Select Identity: Choose the desired identity (sin(A+B), cos(A+B), tan(A+B), sin(A-B), cos(A-B), or tan(A-B)) from the dropdown menu.
- View Results: The calculator will instantly display the primary result (the value of the selected identity), the intermediate trigonometric values of A and B, and the formula used.
- Interpret Chart: For sine and cosine identities, the chart visually represents the components of the formula and the final result.
- Reset or Copy: Use the "Reset" button to clear inputs to default values or "Copy Results" to copy the details to your clipboard.
The angle addition identities calculator provides a quick way to verify manual calculations or find values without manual computation.
Key Factors That Affect Angle Addition Identities Calculator Results
- Values of Angle A and B: The primary inputs that determine all other values.
- Choice of Identity: Selecting sin, cos, or tan, and sum or difference, drastically changes the formula and result.
- Unit of Angles: Our calculator assumes degrees. If your angles are in radians, convert them first (or be aware of the input unit). The underlying `Math.sin`, `cos`, `tan` in JavaScript use radians, so the calculator converts degrees to radians internally.
- Accuracy of sin, cos, tan of A and B: The precision of the individual trig values of A and B affects the final result's accuracy.
- Undefined Tangent Values: For tan(A+B) and tan(A-B), if A, B, A+B, or A-B is 90° + k*180° (where k is an integer), the tangent will be undefined or the denominator will be zero. The angle addition identities calculator attempts to handle these.
- Quadrant of A and B: The signs of sin(A), cos(A), sin(B), cos(B) depend on the quadrants of A and B, which impacts the final result.