Cofunction Calculator
Free Cofunction Calculator for trigonometry. Enter values to get step-by-step solutions with formulas and graphs. Free to use with no signup required.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
sin(x) = cos(90 - x), tan(x) = cot(90 - x), sec(x) = csc(90 - x)
Each trigonometric function of an angle x equals its cofunction evaluated at the complement (90 - x). The three cofunction pairs are sine/cosine, tangent/cotangent, and secant/cosecant.
Worked Examples
Example 1: Cofunction of 30 Degrees
Problem:Verify that sin(30) = cos(60) and tan(30) = cot(60) using cofunction identities.
Solution:The complement of 30 degrees is 90 - 30 = 60 degrees.\nsin(30) = 0.5\ncos(60) = 0.5 (confirmed: sin(30) = cos(60))\ntan(30) = 1/sqrt(3) = 0.57735027\ncot(60) = cos(60)/sin(60) = 0.5/0.86602540 = 0.57735027\nConfirmed: tan(30) = cot(60)
Result:sin(30) = cos(60) = 0.5 | tan(30) = cot(60) = 0.57735027
Example 2: Solving an Equation with Cofunctions
Problem:If sin(x) = cos(35), find the value of x.
Solution:Using the cofunction identity sin(x) = cos(90 - x):\ncos(35) = sin(90 - 35) = sin(55)\nTherefore x = 55 degrees.\nVerification: sin(55) = 0.81915204, cos(35) = 0.81915204.\nThe values match, confirming x = 55 degrees.
Result:x = 55 degrees | sin(55) = cos(35) = 0.81915204
Frequently Asked Questions
What is a cofunction in trigonometry?
A cofunction is a trigonometric function whose value for a given angle equals the value of a complementary trigonometric function evaluated at the complement of that angle. Complementary angles are two angles that sum to 90 degrees. The six standard trigonometric functions form three cofunction pairs: sine and cosine, tangent and cotangent, and secant and cosecant. For example, sin(30) equals cos(60) because 30 and 60 are complementary angles. This relationship arises directly from the geometry of right triangles, where the two acute angles always sum to 90 degrees and the opposite and adjacent sides swap roles when you switch your reference angle.
What are the cofunction identities?
The cofunction identities state that each trigonometric function of an angle equals its cofunction evaluated at the complementary angle. Specifically: sin(x) = cos(90 - x), cos(x) = sin(90 - x), tan(x) = cot(90 - x), cot(x) = tan(90 - x), sec(x) = csc(90 - x), and csc(x) = sec(90 - x). These identities hold for all valid angle values when measured in degrees. In radians, you replace 90 degrees with pi/2, so sin(x) = cos(pi/2 - x) and so forth. These identities are foundational in trigonometry and are frequently used to simplify expressions and solve equations involving complementary angle relationships.
How do you find the cofunction of an angle?
To find the cofunction of an angle, first determine the complement of the angle by subtracting it from 90 degrees (or pi/2 radians). Then evaluate the complementary trigonometric function at that complementary angle. For instance, to find the cofunction of sin(40 degrees), you calculate cos(90 - 40) = cos(50 degrees). Both values will be identical. If you are working with an angle greater than 90 degrees, the cofunction identity still applies algebraically, though the geometric interpretation as complementary angles in a right triangle no longer directly holds. Cofunction Calculator automates the process by computing all six trigonometric functions and their cofunctions simultaneously for any input angle.
What is the difference between a function and its cofunction?
A trigonometric function and its cofunction are distinct functions that happen to produce equal values when their arguments are complementary. Sine measures the ratio of the opposite side to the hypotenuse in a right triangle, while cosine (its cofunction) measures the ratio of the adjacent side to the hypotenuse. When you switch your reference angle to the other acute angle in the triangle, the opposite and adjacent sides swap, which is why sin(x) = cos(90 - x). The prefix co- in cosine, cotangent, and cosecant literally means complement, indicating these functions are the complement-angle counterparts of sine, tangent, and secant respectively. This naming convention dates back to early mathematical terminology.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy