Double Angle Calculator

Doubling an angle does not double its sine, cosine or tangent, and that single fact causes more homework errors than any other trig slip. The double-angle identities give sin(2θ), cos(2θ) and tan(2θ) directly from the functions of θ, and this calculator evaluates all three, computing cos(2θ) three different ways so you can watch the forms agree to eight decimal places.

Enter θ in degrees or radians, or tap a special angle, and the tool shows 2θ in both units plus the full step-by-step substitution. It is a checker for working you have already done, not a replacement for it.

Quick Answer

How to use this calculator

Enter θ, pick degrees or radians, and press Calculate. The result shows 2θ in both units, sin(2θ), cos(2θ) in all three forms, tan(2θ), and the original sin(θ), cos(θ) and tan(θ) for comparison, followed by the substitution lines. The quick picks (30°, 45°, 60°, 90°) calculate immediately, which makes them handy for checking a textbook table line by line. Results stay hidden until you calculate, and Reset clears the angle back to blank.

Use it as a homework checker, not a homework replacement. Work the identity by hand first, then compare: the tool shows eight decimals, so a sign error or a swapped form stands out at once. Two habits make the check meaningful. First, predict before you click: write down what you expect each form to give, then look for the one that disagrees with your prediction rather than scanning for agreement. Second, test an ugly angle (θ = 37°, say): round numbers can hide sign errors, because sin(60°) looks right even when your working is wrong.

Worked example, checked against this exact tool

θ = 30°, so 2θ = 60°. sin(60°) = 0.86602540; cos(60°) = 0.50000000 via all three forms (cos²−sin², 2cos²−1, 1−2sin²); tan(60°) = 1.73205081. Original values: sin(30°) = 0.50000000, cos(30°) = 0.86602540, tan(30°) = 0.57735027. The tool prints the agreement line underneath: all three cosine forms agree, so the identity holds for this angle.

An ugly-angle check: θ = 37° gives 2θ = 74°, sin(74°) = 0.96126170, cos(74°) = 0.27563736 via all three forms, and tan(74°) = 3.48741444. If your hand calculation lands on the same eight decimals for 37°, the working is solid in a way that a 30° check alone cannot prove.

The identities

There are five formulas to learn, but only three ideas. Sine has one form: sin(2θ) = 2 sin θ cos θ. Cosine has three equivalent forms: cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. Tangent has one: tan(2θ) = 2 tan θ ÷ (1 − tan²θ). Pick the cosine form that matches what you already know: if you only know sin θ, use 1 − 2sin²θ; if you only know cos θ, use 2cos²θ − 1; if you know both, the first form is the most direct.

If you need sin, cos and tan of θ itself rather than of 2θ, our trigonometry angle calculator evaluates those directly, and our reference angle calculator reduces any angle to its acute reference angle first.

Exact values worth knowing

For the standard angles, the double-angle values have clean exact forms. The decimals below are those exact values rounded to eight places for display: the identities themselves are exact, and only the display is rounded.

θ	2θ	sin(2θ)	cos(2θ)	tan(2θ)
0°	0°	0	1	0
30°	60°	√3/2 ≈ 0.86602540	1/2	√3 ≈ 1.73205081
45°	90°	1	0	undefined
60°	120°	√3/2 ≈ 0.86602540	−1/2	−√3 ≈ −1.73205081
90°	180°	0	−1	0

Where the identities come from

They are the angle-sum identities with both angles set equal. Start from sin(A + B) = sin A cos B + cos A sin B and set B = A: sin(2θ) = 2 sin θ cos θ. Likewise cos(A + B) = cos A cos B − sin A sin B becomes cos(2θ) = cos²θ − sin²θ, and the other two cosine forms follow from sin²θ + cos²θ = 1: replace cos²θ with 1 − sin²θ for the third form, and sin²θ with 1 − cos²θ for the second. The tangent form is sin(2θ) divided by cos(2θ), simplified. Nothing here is a definition to memorise blindly: each line is one substitution away from the sum formulas.

When tangent is undefined, and when it only looks huge

tan(2θ) is undefined where its denominator vanishes: 2θ = 90° + k·180°, that is, θ = 45° + k·90°. At θ = 45°, for example, tan(90°) has no finite value and the tool says so instead of printing a huge number. The tool decides this with a threshold: it prints "undefined" when the absolute value of 1 − tan²θ falls below 0.0000000001.

Just outside that threshold, the tool prints a very large finite number, and that number is real behaviour, not an error to fix: the tangent function blows up near its asymptote. At θ = 44.99°, tan(2θ) prints as 2864.78885930, because 2θ = 89.98° sits just short of 90°. Read such a value as "close to the asymptote, growing without bound", not as a usable angle answer. The same caution applies at θ = 90° itself in the other direction: tan(θ) is undefined there, and the tool's "Original" line prints a floating-point artifact (a huge finite number) for tan(90°), while tan(180°) = −0.00000000, which is 0, is correct. Trust the 2θ line at 90°; ignore the original tan line at exactly 90°.

Degrees vs radians

Enter θ in whichever unit you have; the tool converts internally and reports 2θ in both. The identities are unit-agnostic: 30° and π/6 radians give identical results (sin 0.86602540, cos 0.50000000, tan 1.73205081), and the tool's step-by-step line records which unit you entered. Our triangle formula calculator handles sides, angles and area in one place when the problem is a triangle rather than an identity.

A numbered walkthrough: θ = 45°

Setting θ = 45° makes 2θ = 90°, a case where every identity should reproduce a result you already know. Here is the substitution, line by line:

1. sin(90°) = 2 sin 45° cos 45° = 2 × (√2/2) × (√2/2) = 2 × 1/2 = 1. Correct.
2. cos(90°) three ways: cos²45° − sin²45° = 1/2 − 1/2 = 0; 2cos²45° − 1 = 2(1/2) − 1 = 0; 1 − 2sin²45° = 1 − 2(1/2) = 0. All three agree: 0.
3. tan(90°) = 2tan45° ÷ (1 − tan²45°) = 2 ÷ (1 − 1): undefined. The tool reports "undefined" here instead of printing a huge number, which is the honest answer.

Run the same angle through the tool above and compare each line. If the three cosine forms ever disagreed, that would mean an arithmetic slip, not a broken identity: they are the same statement in three forms. An earlier version of this page claimed the tool "flags" a disagreement, but the old code only printed the agreement line and stayed silent otherwise. The tool above now prints an explicit disagreement warning if the forms ever differ by more than 0.000000001, so silence can no longer be mistaken for agreement.

Solving an equation with the identities

Double-angle identities turn trigonometric equations into ordinary quadratics you already know how to solve. The pattern is always the same: replace, substitute, factor. Take sin(2θ) = 1/2 for θ between 0° and 180°. Reading the identity the other way gives 2θ = 30° or 150° (the two angles in range with sine 1/2), so θ = 15° or 75°. Two solutions, both checkable in the tool.

A meatier one: solve cos(2θ) = cos θ. Pick the cosine form that matches what is already there: 2cos²θ − 1 = cos θ, so 2cos²θ − cos θ − 1 = 0. Factor: (2cos θ + 1)(cos θ − 1) = 0, giving cos θ = −1/2 or cos θ = 1. In [0°, 360°) that is θ = 120°, 240°, or 0°. The identity did the hard part; the quadratic finished it, and our quadratic equation calculator solves that last step with the working shown.

Power-reduction: the identities running backward

Read the cosine forms right to left and they become power-reduction formulas: sin²θ = (1 − cos 2θ)/2 and cos²θ = (1 + cos 2θ)/2. They trade a squared trig function for a first-power one, which is exactly the trade calculus wants. This is how you integrate sin²x without a trick: replace it with (1 − cos 2x)/2 and integrate term by term. It is also the math behind average AC power: the mean of sin² over a cycle is 1/2, which falls straight out of the formula. Replace θ with θ/2 and you get the half-angle formulas for free: sin²(θ/2) = (1 − cos θ)/2, with the sign of sin(θ/2) itself taken from the quadrant.

Outside class, the same identities do working jobs. Physics uses sin(2θ) for projectile range: maximum range at a 45° launch falls out of the identity. Electrical engineering leans on power-reduction to compute average AC power from sin² waveforms. Calculus uses both directions: double-angle to build integrands you can handle, power-reduction to integrate sin² and cos². They are working tools, not just exam material.

What competing calculators miss

Most double-angle tools evaluate the identities and stop, often showing only one cosine form, and some give reciprocals (cosecant, secant, cotangent) without the step-by-step substitution that shows where the numbers came from. A bare evaluator also hides the interesting cases: it prints a huge tangent near 45° with no note that the function is blowing up, and it never shows whether its own cosine forms agree. This page keeps all three cosine forms visible with an explicit agreement check, shows the original θ values beside the doubled ones, records the substitution steps, and says plainly where the undefined threshold sits and what the 90° artifact line means. The reciprocals are deliberately left out: csc(2θ) = 1 ÷ sin(2θ), sec(2θ) = 1 ÷ cos(2θ) and cot(2θ) = (1 − tan²θ) ÷ (2 tan θ) follow at once from the three results above, and adding three more outputs would bury the pedagogical point, which is watching the three cosine forms agree.

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