Library / Geometry, coordinates, surveying and trigonometry / GEO-41
Geometry, coordinates, surveying and trigonometry
Tangent addition identity
Tangent of the sum of two angles from the tangents u = tan α and v = tan β.
- Ref
- GEO-41
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- \tan(α+β)
The tangent of a sum is tan(α + β) = (tan α + tan β)/(1 − tan α tan β). It comes from dividing the sum formulas for sine and cosine by cos α cos β. If uv = 1 the denominator is zero, which means α + β is a quarter turn and its tangent is undefined. Values near that pole are very sensitive to rounding.
Tangent of the sum of two angles from the tangents u = tan α and v = tan β. The product uv must not be 1, since the tangent is then undefined, and values near it are sensitive to rounding.
Calculator
- Sum of the angles40.6 °
- Denominator, 1 − uv0.875
- First angle, arctan u26.57 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Angles with tangents 0.5 (26.57°) and 0.25 (14.04°) add to 40.6°, whose tangent is 0.8571.
- Tangents of 1/2 and 1/3 give exactly 1, so arctan(1/2) + arctan(1/3) = 45°, a well-known check.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| u Tangent of the first angle, tan α | −5 to 3.9 | −2.111 | 166 | 168.1 | +0.17800000000000002 | +30.4 % |
| v Tangent of the second angle, tan β | −5 to 1.9 | −1.286 | 48 | 49.29 | +0.138 | +28.5 % |
Sweep of Tangent of the first angle, tan α
| Tangent of the first angle, tan α | Tangent of the sum |
|---|---|
| −5 | −2.111 |
| −3.888 | −1.845 |
| −2.775 | −1.491 |
| −1.662 | −0.9978 |
| −0.55 | −0.2637 |
| 0.5625 | 0.9455 |
| 1.675 | 3.312 |
| 2.788 | 10.02 |
| 3.9 | 166 |
Sweep one variable
Variables and how each one contributes
Tangent of the sum
What the equation gives you. Everything else feeds it.
Tangent of the first angle, tan α
The tangent of the first angle, tan α, a plain number. It must be finite. Raises the tangent of the sum, with larger effect as uv approaches 1. Raising it by 0.1 (to 0.6) moves the result by +16.7 %, from 0.8571 to 1.
Tangent of the second angle, tan β
The tangent of the second angle, tan β, a plain number. It must be finite. Raises the tangent of the sum, with larger effect as uv approaches 1. Raising it by 0.1 (to 0.35) moves the result by +20.2 %, from 0.8571 to 1.03.
Worked example with the values above
- Numerator: u + v = 0.5 + 0.25 = 0.75.
- Denominator: 1 − uv = 1 − 0.125 = 0.875.
- tan(α + β) = 0.85714.
Assumptions and limits
- The inputs are the exact sine, cosine or tangent values of a real angle.
- Where a square root is taken, the stated range of the angle fixes the sign.
Common mistakes
- Using the formula when the tangent is undefined, such as at 90°.
- Taking the positive square root outside the range where the sign is positive.
- Mixing degrees and radians when using a calculator for the final angle.
Statistics and uncertainty
Identities are exact, so the only error is numerical. Tangent values near a pole (uv close to 1) magnify rounding, and a square root of a number near zero loses digits.
Where it is used
- Simplifying expressions and solving trigonometric equations in mathematics and physics.
- Combining two sinusoids of the same frequency into one wave, as in phasors, AC circuits and vibration.
Maths you need
- Sine, cosine and tangent: ratios from a right triangle or the unit circle
- Pythagorean identity: sin² + cos² = 1
- Phase: the horizontal shift of a wave