Mathematics
Trigonometric formulas calculator
Enter one angle — and a second one if you need it — and see every identity with its formula beside the value: half-angle, double-angle, addition, prosthaphaeresis and Werner.
How these formulas are used
Trigonometric formulas exist to rewrite an expression into a more workable form, not to produce a number: any phone will give you the number. The value beside each identity is therefore a check — it tells you the rewriting holds for your angles — while the formula is what you carry into the exercise.
The half-angle formulas express the sine, cosine and tangent of α/2 using only the cosine of α. The classic form carries a ± in front of the radical, and choosing the sign depends on the quadrant α/2 falls in: it is the most common mistake in exercises. Here the value is computed directly on α/2, so the sign is already right and you can use it to check your own choice.
The prosthaphaeresis formulas turn sums into products, and they are at their most useful when solving an equation: a sum of sines does not vanish easily, a product does, because one factor going to zero is enough. The Werner formulas go the other way and are what you need when a product has to be integrated or simplified.
Common mistakes
- Getting the sign wrong in the half-angle formulas: the ± is not a free choice, the quadrant of α/2 decides it — not the quadrant of α.
- Feeding degrees into a formula that assumes radians: check the selected unit before reading the values.
- Forgetting that tangent and cotangent do not exist everywhere: tan α is undefined at 90°, cot α at 0°. Where that happens the table says so instead of showing a huge number.
Frequently asked questions
What are the prosthaphaeresis formulas?
They are the four identities that turn a sum or difference of sines and cosines into a product: for instance sin α + sin β = 2·sin((α+β)/2)·cos((α−β)/2). They matter most in equations, because a product vanishes as soon as one factor does.
Why do the half-angle formulas carry a double sign?
Because the radical always returns a positive value, while the sine and cosine of α/2 can be negative. The quadrant α/2 falls in decides the sign: for α = 300°, for example, α/2 = 150° lands in the second quadrant, where the cosine is negative.
Can I work in radians?
Yes, switch the unit in the selector. The calculator shows both readings of the angle anyway, so you can confirm you picked the unit you meant.
How this calculation works
Half-angle: sin(α/2) = ±√((1−cos α)/2), cos(α/2) = ±√((1+cos α)/2), tan(α/2) = sin α/(1+cos α). Double angle: sin 2α = 2 sin α cos α, cos 2α = cos²α − sin²α, tan 2α = 2 tan α/(1−tan²α). Addition: sin(α±β) = sin α cos β ± cos α sin β, cos(α±β) = cos α cos β ∓ sin α sin β. Prosthaphaeresis: sin α + sin β = 2 sin((α+β)/2) cos((α−β)/2); sin α − sin β = 2 cos((α+β)/2) sin((α−β)/2); cos α + cos β = 2 cos((α+β)/2) cos((α−β)/2); cos α − cos β = −2 sin((α+β)/2) sin((α−β)/2). Werner: sin α sin β = ½[cos(α−β) − cos(α+β)]; cos α cos β = ½[cos(α−β) + cos(α+β)]; sin α cos β = ½[sin(α+β) + sin(α−β)].