Math
Double Angle Formula Calculator
The double-angle formulas express sin 2θ, cos 2θ, and tan 2θ in terms of sin θ, cos θ, and tan θ.
Give the calculator the angle, or give it one ratio such as sin θ = 3/5 with the quadrant, and it completes the other ratios, applies each formula with exact fractions or radicals, and shows all three forms of cos 2θ.
Fractions, decimals, and radicals work. Keep sin and cos between -1 and 1.
The quadrant decides the sign of the ratios you were not given.
Try an example
Result
sin 2θ
—
- cos 2θ
- —
- tan 2θ
- —
- sin θ
- —
- cos θ
- —
- tan θ
- —
- cos 2θ, three forms
- —
More details (1 more)
- 2θ
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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The double-angle identities
They come from the sum identities with α = β = θ. The three cos 2θ forms are the same identity rewritten with sin²θ + cos²θ = 1, and you choose whichever uses the ratio you know.
Solving from one ratio and a quadrant
- Use sin²θ + cos²θ = 1 (or 1 + tan²θ = sec²θ) to find the missing ratio's size.
- Use the One of the four regions the x- and y-axes cut the coordinate plane into. to fix its sign: sine is positive in Quadrants I and II, cosine in I and IV, tangent in I and III.
- Substitute into the double-angle formulas.
sin θ = 3/5 in Quadrant II
cos θ = -√(1 − 9/25) = -4/5. sin 2θ = 2(3/5)(-4/5) = -24/25. cos 2θ = 16/25 − 9/25 = 7/25. tan 2θ = -24/7.
How to read the answer
- Exact answers appear as fractions or A root expression written with the √ symbol, such as a square root. with the decimal in parentheses.
- tan 2θ is undefined when cos 2θ = 0, which happens when tan θ = ±1 (θ = 45°, 135°, ...).
- 2θ may land in a different quadrant than θ; the calculator reports it so you can sanity-check the signs.
Common mistakes and edge cases
- sin 2θ is not 2 sin θ. Doubling the angle does not double the sine.
- Forgetting the quadrant sign when taking the square root for the missing ratio.
- Entering sin θ = 0.6 in Quadrant III: sine is negative there, so the input is inconsistent and the calculator says so.
- Using tan 2θ = 2 tan θ / (1 − tan²θ) when tan θ = ±1: the The bottom number of a fraction; it says how many equal parts make one whole. is zero, so tan 2θ is undefined.
Worked examples
sin θ = 3/5 in Quadrant II
sin 2θ
-24/25 (-0.96)
cos θ = 5/13 in Quadrant IV
sin 2θ
-120/169 (-0.710059)
tan θ = 4/3 in Quadrant III
sin 2θ
24/25 (0.96)
From the angle
θ = 30° gives sin 60°
sin 2θ
√3/2 (0.866025)
Undefined tan 2θ
θ = 45°, so 2θ = 90°
sin 2θ
1
Radical input
sin θ = √3/2 in Quadrant I
sin 2θ
√3/2 (0.866025)
Sign conflict
sin θ = 0.6 cannot be in Quadrant III
sin 2θ
—
Impossible ratio
sin 2θ
—
Frequently asked questions
What is the double angle formula for sine?+
sin 2θ = 2 sin θ cos θ. For θ = 30°: 2 × 1/2 × √3/2 = √3/2, which is sin 60°.
Why does cos 2θ have three forms?+
Start with cos²θ − sin²θ and replace sin²θ with 1 − cos²θ to get 2cos²θ − 1, or replace cos²θ with 1 − sin²θ to get 1 − 2sin²θ. Use the form that matches the ratio you know.
How do I find sin 2θ if I only know sin θ?+
Find cos θ from cos²θ = 1 − sin²θ, choose its sign from the quadrant, then multiply: sin 2θ = 2 sin θ cos θ.
When is tan 2θ undefined?+
When cos 2θ = 0, which happens for θ = 45° + 90°k. In the formula 2 tan θ / (1 − tan²θ) this shows up as a zero denominator because tan θ = ±1.
Do the formulas work in radians?+
Yes. They are identities in the angle, whatever unit you use. Enter θ = π/6 in radian mode and you get sin 2θ = sin(π/3) = √3/2.
Is sin 2θ the same as 2 sin θ?+
No. For θ = 90°, sin 2θ = sin 180° = 0 while 2 sin θ = 2. The double-angle formula is the correct expansion.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026