Calculator Form
Plotly Graph
The graph follows the selected trigonometric function and highlights the combined angle from your current result.
Example Data Table
| Case | Identity | Angles | Exact Result | Decimal Result |
|---|---|---|---|---|
| Example 1 | sin(A + B) | 45° and 30° | (√6 + √2)/4 | 0.965926 |
| Example 2 | cos(A - B) | 75° and 30° | √2/2 | 0.707107 |
| Example 3 | tan(A - B) | 45° and 30° | 2 - √3 | 0.267949 |
| Example 4 | cos(A + B) | 60° and 45° | (√2 - √6)/4 | -0.258819 |
| Example 5 | sin(A - B) | 75° and 45° | 1/2 | 0.500000 |
Formulas Used
Sine sum: sin(α + β) = sin α cos β + cos α sin β
Sine difference: sin(α - β) = sin α cos β - cos α sin β
Cosine sum: cos(α + β) = cos α cos β - sin α sin β
Cosine difference: cos(α - β) = cos α cos β + sin α sin β
Tangent sum: tan(α + β) = (tan α + tan β) / (1 - tan α tan β)
Tangent difference: tan(α - β) = (tan α - tan β) / (1 + tan α tan β)
Exact values are shown whenever the final angle matches a built-in special angle such as 15°, 30°, 45°, 60°, 75°, or their quadrant reflections.
How to Use This Calculator
- Select the identity you want to evaluate.
- Choose preset exact angles for symbolic work, or custom numeric angles for direct calculation.
- Pick degrees or radians.
- Set your preferred decimal precision and graph range.
- Press Calculate to show the result above the form.
- Review the identity, exact substitution, decimal verification, and graph marker.
- Use the CSV or PDF buttons to save the result.
FAQs
What do sum and difference formulas do?
They rewrite sine, cosine, or tangent of combined angles into separate angle parts. This makes exact values easier when the two angles already have known trigonometric values, such as 45° and 30°.
Which identities are included here?
The calculator handles sin(A+B), sin(A-B), cos(A+B), cos(A-B), tan(A+B), and tan(A-B). It shows the identity used, the combined angle, the exact form for common angles, and a decimal check.
Use a sum or difference formula to find the exact value
Choose preset angles with known exact trigonometric values, such as 45° and 30°. Pick the needed identity, submit, and simplify the substitution shown. For example, sin(45°+30°) becomes sin45°cos30° + cos45°sin30°.
Can I use radians instead of degrees?
Yes. Switch the unit to radians and enter custom values or choose preset angles labeled in π notation. The calculator evaluates the same identities and still gives decimal verification from radian input.
Why are some tangent results undefined?
Tangent is undefined whenever cosine equals zero, such as π/2 or 90°. Near those angles, the graph rises sharply and the calculator flags the result instead of forcing a misleading number.
Why do exact answers contain radicals?
Common special angles have exact sine and cosine values involving square roots. When identities combine them, radicals remain. Those forms are exact and preferred in proofs, symbolic work, and exam solutions.
Do custom angles also return exact symbolic answers?
Custom numeric angles always return a decimal result. Exact symbolic output appears only when the calculator can match common special angles reliably. That keeps the algebra accurate and avoids false simplifications.
How can this help with homework or revision?
It lets you compare the direct value with the expanded identity. That is useful for checking algebra, signs, quadrants, and final answers before submitting homework or revision notes.