What are cofunction identities in trigonometry?

Cofunction identities state that each trigonometric function of an angle equals its paired cofunction evaluated at the complementary angle (90 degrees minus the angle). The three pairs are: sine and cosine (sin theta = cos(90 minus theta)), tangent and cotangent (tan theta = cot(90 minus theta)), and secant and cosecant (sec theta = csc(90 minus theta)).

What is the cofunction of sin(30)?

The cofunction of sin(30 degrees) is cos(60 degrees). Since 30 and 60 are complementary angles (they add to 90), sin(30) = cos(60) = 0.5. This is the cofunction identity for sine and cosine: sin(theta) = cos(90 minus theta).

Why are cofunctions called co-functions?

The prefix co- in cosine, cotangent and cosecant comes from the word complement. Cosine literally means complement of sine: it gives the sine of the complementary angle. The same logic applies to cotangent (complement of tangent) and cosecant (complement of secant). This naming reflects the fundamental cofunction identities.

Cofunction Calculator

The cofunction identities are among the most elegant relationships in trigonometry. They state that every trigonometric function of an angle is equal to its paired cofunction evaluated at the complementary angle, which is 90 degrees minus the original angle. The three cofunction pairs are sine and cosine, tangent and cotangent, and secant and cosecant. The prefix co- in cosine, cotangent and cosecant literally refers to complement: cosine is the sine of the complementary angle. So if you know sin(30) = 0.5, you automatically know that cos(60) = 0.5 without any extra work, because 30 and 60 are complementary. The same applies to all six trig functions. This is useful in simplifying expressions, proving trig identities, and working with right-angled triangles where the two acute angles always add up to 90 degrees. This calculator takes any angle in degrees and returns all six trig values at that angle and all six at the complementary angle. It is set up so you can verify the cofunction identity visually: each value in the left column should match its cofunction in the right column. The tool handles angles outside 0 to 90 degrees too, where the cofunction identity still holds algebraically. If a function is undefined at your chosen angle (tangent at 90 degrees, for example), the calculator shows that clearly.

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Angle (degrees)

deg

0.5000

sin(30) = cos(60)

Complement60°

cos(30)0.8660

tan(30)0.5774

cot(30)1.7321

sec(30)1.1547

csc(30)2.0000

Cofunction identities: sin(t)=cos(90-t), cos(t)=sin(90-t), tan(t)=cot(90-t), cot(t)=tan(90-t), sec(t)=csc(90-t), csc(t)=sec(90-t).

Function	At θ	Cofunction	At (90 - θ)
sin(θ)	0.5000	cos(90-θ)	0.5000
cos(θ)	0.8660	sin(90-θ)	0.8660
tan(θ)	0.5774	cot(90-θ)	0.5774
cot(θ)	1.7321	tan(90-θ)	1.7321
sec(θ)	1.1547	csc(90-θ)	1.1547
csc(θ)	2.0000	sec(90-θ)	2.0000

How it works

For any angle theta in degrees, the complement is (90 - theta). The cofunction identities are:
sin(theta) = cos(90 - theta)
cos(theta) = sin(90 - theta)
tan(theta) = cot(90 - theta)
cot(theta) = tan(90 - theta)
sec(theta) = csc(90 - theta)
csc(theta) = sec(90 - theta)
These hold for all angles where the functions are defined. In this calculator, angles are converted to radians before applying JavaScript's built-in Math functions. Cotangent, secant and cosecant are derived as 1/tan, 1/cos and 1/sin respectively.

Worked example

With angle = 30 degrees: complement = 60 degrees. sin(30) = 0.5000 and cos(60) = 0.5000, confirming the identity. cos(30) = 0.8660 and sin(60) = 0.8660. tan(30) = 0.5774, cot(30) = 1.7321, sec(30) = 1.1547, csc(30) = 2.0000. Each value at 30 degrees equals its cofunction pair at 60 degrees.

Related calculators

Coterminal Angle Calculator: find positive and negative coterminal angles for any input.
Trig Identity Calculator: verify the major trig identities numerically at any angle.
Law of Sines and Cosines: solve triangle sides and angles using trig laws.
Percent Error Calculator: compare your computed trig value to a reference.

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