Compute sin(2θ), cos(2θ), and tan(2θ) with double-angle identities from an angle, known sine, known cosine, or known tangent plus quadrant.
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Double-angle identities Enter an angle or a known trigonometric value to compute sin(2θ), cos(2θ), and tan(2θ), with the identity forms shown for checking algebra and homework steps.
Quick presets
Input note
Enter θ directly in degrees or radians.
Results
sin(2θ) = 0.8660254038
2sin(θ)cos(θ) using sin(θ) = 0.5 and cos(θ) = 0.8660254038.
Double-angle calculator for sin(2θ), cos(2θ), and tan(2θ)
The double-angle calculator computes sin(2θ), cos(2θ), and tan(2θ) from an angle in degrees or radians, or from a known sine, cosine, or tangent value plus the quadrant for θ. It is built for trigonometry identity checks, exact-value homework, calculus reduction formulas, and practical angle-doubling problems where the sign and undefined tangent cases matter.
The double-angle formulas
Double-angle identities express the trig functions of twice an angle in terms of the original angle. The three core formulas are sin(2θ) = 2 sin(θ) cos(θ), cos(2θ) = cos²(θ) - sin²(θ), and tan(2θ) = 2 tan(θ) / (1 - tan²(θ)).
The cosine double-angle formula has three equivalent forms. Depending on the information you know, cos(2θ) can be written as cos²(θ) - sin²(θ), 2cos²(θ) - 1, or 1 - 2sin²(θ). The calculator shows all three forms side by side so you can choose the shortest path for the problem in front of you.
sin(2θ) = 2sin(θ)cos(θ)
Use when you know both sine and cosine of the original angle.
Use when tangent is known and the denominator is not zero.
Using an angle in degrees or radians
If you know the angle θ, enter it directly and choose degrees or radians before reading the result. The calculator doubles the angle, evaluates the original sine, cosine, and tangent values, and then reports sin(2θ), cos(2θ), and tan(2θ) with the doubled angle shown in both units.
This angle-first workflow covers searches such as double angle calculator, sin 2x calculator, cos 2x calculator, and tan 2x calculator. It also helps catch the common mistake of treating sin(2x) as the same thing as 2sin(x), which is not true because sine is not a linear function.
Using a known sine, cosine, or tangent value
Many textbook problems give information like sin(θ) = 3/5 with θ in quadrant I, cos(θ) = -4/5 with θ in quadrant II, or tan(θ) = -3/4 with θ in quadrant IV. In those cases, the angle itself is not necessary. The calculator reconstructs the missing trig values from the Pythagorean identity and the selected quadrant, then applies the double-angle formulas.
This is where the quadrant selector matters. A sine value alone may match two different quadrants, and a tangent value repeats every 180 degrees. The sign of the recovered sine or cosine can change the signs of sin(2θ) and tan(2θ), so the calculator rejects input values whose sign conflicts with the quadrant instead of silently producing a misleading answer.
Choosing the right cosine form
Use cos(2θ) = cos²(θ) - sin²(θ) when both sine and cosine are already known. Use cos(2θ) = 2cos²(θ) - 1 when cosine is the cleanest input. Use cos(2θ) = 1 - 2sin²(θ) when sine is the cleanest input.
Competitor calculators often hide this decision by returning only one decimal result. Calcipedia keeps the identity check visible because it is useful for proof work, integration, and simplifying trigonometric expressions where the best formula depends on the expression you are trying to rewrite.
When tan(2θ) is undefined
The tangent double-angle formula has a denominator of 1 - tan²(θ). If tan(θ) = 1 or tan(θ) = -1, the denominator is zero, so tan(2θ) is undefined. The same idea appears when 2θ lands at an angle where cosine is zero, such as 90 degrees or 270 degrees.
The calculator labels undefined tangent values explicitly. That warning is more useful than returning 0, Infinity, or a rounded floating-point artifact because the undefined state is part of the mathematical answer.
When to use double-angle identities
Double-angle formulas simplify expressions in algebra and precalculus, reduce powers in calculus integrals, support wave and signal calculations in physics and engineering, and help derive half-angle identities. They are also common in exact-value problems where you know a special angle or a simple trig ratio.
Use this page when the problem asks for sin(2θ), cos(2θ), tan(2θ), sin(2x), cos(2x), tan(2x), or a double-angle identity check. Use the broader trigonometry calculator when the problem is about evaluating ordinary trig functions, inverse trig, reference angles, or unit-circle lookup.
A double-angle calculator finds sin(2θ), cos(2θ), and tan(2θ) from information about θ. This page accepts an angle in degrees or radians, or a known sine, cosine, or tangent value with a quadrant.
What is sin(2 × 30°)?
sin(60°) = sqrt(3)/2, which is about 0.866. Using the formula, 2 sin(30°) cos(30°) = 2 × 0.5 × sqrt(3)/2 = sqrt(3)/2.
Is sin(2x) the same as 2sin(x)?
No. sin(2x) means the sine of a doubled angle. 2sin(x) means twice the sine value. For example, if x = 30 degrees, sin(60 degrees) is about 0.866, while 2sin(30 degrees) equals 1.
Which cosine double-angle formula should I use?
Use cos²θ - sin²θ when both values are known, 2cos²θ - 1 when cosine is known, and 1 - 2sin²θ when sine is known. The calculator shows all three forms so you can match the form to the information you have.
Why does the calculator ask for a quadrant when I enter sin θ, cos θ, or tan θ?
The quadrant determines the sign of the missing trig value. For example, sin θ = 3/5 can belong to quadrant I or II, but cos θ is positive in quadrant I and negative in quadrant II, which changes the double-angle result.
When is tan(2θ) undefined?
tan(2θ) is undefined when cos(2θ) = 0. In the tangent formula, that shows up when 1 - tan²(θ) = 0, or tan(θ) = ±1.
Can I calculate double-angle values from tan θ only?
Yes. Enter tan θ and choose the quadrant. The calculator reconstructs compatible sine and cosine values, computes sin(2θ), cos(2θ), and then checks whether tan(2θ) is defined.
Should I use degrees or radians?
Use the unit required by your problem. Geometry and many classroom examples use degrees, while calculus, graphing, physics, and signal problems often use radians. The angle workflow shows the doubled angle in both units.
How can I check the double-angle identities manually?
First compute sin θ, cos θ, and tan θ for the original angle or ratio. Then substitute those values into sin(2θ) = 2sinθcosθ, one of the cosine forms, and tan(2θ) = 2tanθ / (1 - tan²θ). Compare each step with the calculator output.
Does this replace a symbolic identity prover?
No. This calculator evaluates numeric double-angle identities and shows the main formula forms. Use a symbolic algebra system when you need to prove a complicated identity with variables only.
