Math
Cofunction Calculator
Cofunctions are pairs of trig functions that swap when the angle is replaced by its complement: sine and cosine, tangent and cotangent, secant and cosecant.
Pick a function and an angle, and the calculator writes the identity, evaluates both sides with exact values where possible, and lists every pair.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
Cofunction identity
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- Value of the original function
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- Value of the cofunction
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- Complementary angle
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- Cofunction pair
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- All six pairs at this angle
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The cofunction identities
Why they work
In a right triangle the two acute angles add to 90°. The side opposite one angle is adjacent to the other, so opposite/The longest side of a right triangle, always opposite the right angle. for the first angle equals adjacent/hypotenuse for the second. That is sin θ = cos(90° − θ), and the other pairs follow the same way. The prefix co- literally means complement.
How to use it
- Select the function you have and enter the angle in degrees or An angle unit where a full turn is 2π instead of 360 degrees..
- Read the rewritten identity in the header, then compare the two values below it.
- Use the all-six-pairs line when a problem asks you to convert several functions at once.
How to read the answer
- The two values are equal every time; that is the identity. The answer kept as a fraction or radical instead of a rounded decimal. are shown for special angles.
- For angles outside 0° to 90° the complement is negative or larger than 90°, which is fine algebraically but no longer describes a right triangle.
- In radian mode the complement is shown as π/2 − θ with an exact π fraction when possible.
Common mistakes and edge cases
- Using 180° − θ: that is the supplement and gives sin θ = sin(180° − θ), a different identity.
- Pairing sine with tangent: cofunctions are sin/cos, tan/cot, and sec/csc only.
- Mixing units: in radians subtract from π/2 ≈ 1.5708, not from 90.
- At θ = 0° the cofunction of cot is tan 90°, which is undefined; both sides are undefined together.
Worked examples
sin 35°
Cofunction identity
sin 35° = cos(90° − 35°) = cos 55°
tan 30° exactly
tan 30° = cot 60° = √3/3
Cofunction identity
tan 30° = cot(90° − 30°) = cot 60°
Radians
cos(π/6) = sin(π/3)
Cofunction identity
cos(π/6) = sin(π/2 − π/6) = sin(π/3)
Secant
sec 20° = csc 70°
Cofunction identity
sec 20° = csc(90° − 20°) = csc 70°
Obtuse angle
sin 120° = cos(-30°)
Cofunction identity
sin 120° = cos(90° − 120°) = cos -30°
Zero angle
cot 0° and tan 90° are both undefined
Cofunction identity
cot 0° = tan(90° − 0°) = tan 90°
Unreadable input
Cofunction identity
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Frequently asked questions
What is a cofunction?+
A function whose value at the complement of an angle equals the original function at the angle. Cosine is the cofunction of sine, cotangent of tangent, cosecant of secant.
Why does sin 30° equal cos 60°?+
Because 30° and 60° are complementary. In a 30-60-90 triangle the side opposite 30° is the side adjacent to 60°, so the two ratios are the same number, 1/2.
What is the cofunction identity in radians?+
Replace 90° with π/2: sin θ = cos(π/2 − θ), tan θ = cot(π/2 − θ), sec θ = csc(π/2 − θ).
Do cofunction identities work for angles bigger than 90°?+
Yes, for every angle. sin 120° = cos(90° − 120°) = cos(-30°) = √3/2. The complement is just no longer an acute angle.
How do I solve sin(2x + 10°) = cos(3x − 20°)?+
Use the cofunction identity: the arguments must be complementary, so 2x + 10° + 3x − 20° = 90°, giving 5x = 100° and x = 20° (plus periodic solutions).
Is cos the cofunction of sin or the reciprocal?+
The cofunction. The reciprocal of sine is cosecant (1 / sin θ). Cofunction and reciprocal are different relationships.
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