Cofunction Calculator

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Cofunction

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What is Cofunction Calculator?

Cofunction identities let you rewrite a trigonometric function using a complementary angle. For example, the sine of 30 degrees equals the cosine of 60 degrees. The angle changes, and the function changes with it, but the value stays the same wherever the expressions are defined.

This calculator performs that symbolic rewrite for sine, cosine, tangent, cotangent, secant, and cosecant. It returns an expression such as cos(60°), not a decimal evaluation. That makes it useful when an exercise asks for an equivalent expression or when you want to recognize two apparently different trig terms.

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How to Use This Calculator

Select the original function in the Function menu. Enter the angle as a number of degrees in Angle (x), then select Calculate. To rewrite sin(30°), choose sin(x) and enter 30. The result is cos(60°).

The form has no radians selector. If your problem gives π/6 radians, convert it to 30 degrees before entering it. Do not type the symbols π or an algebraic expression into the numeric field. Negative and greater-than-90-degree angles can be rewritten algebraically, but their complements are no longer both acute angles inside a right triangle.

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Formula Used

The complementary angle is 90° − x. The six identities are 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).

For radians, replace 90 degrees with π/2; that is a mathematical identity, not a second input mode in this form. The pairings follow the opposite and adjacent sides of complementary acute angles in a right triangle. Switching which acute angle you consider swaps those sides while leaving the hypotenuse unchanged.

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Variables Explained

Symbol or inputMeaning
FunctionThe starting trigonometric function. Its cofunction is chosen from the three pairs: sine/cosine, tangent/cotangent, secant/cosecant.
xOriginal angle entered in degrees. It is not a function value or a side length.
90° − xComplementary angle used in the rewritten expression. It can be zero, negative, or greater than 90 degrees.
ResultA symbolic expression. The calculator does not evaluate its decimal value or check every domain restriction.
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Worked Example

Start with sin(30°). Subtract 30 from 90 to get 60, then replace sine with cosine. The result is cos(60°). Both values are 1/2, so the rewrite preserves the original value.

For tan(25°), the complementary angle is 65 degrees, and the result is cot(65°). You do not take the reciprocal of tan(25°); that would be cot(25°), a different expression. The reciprocal relationship uses the same angle, while a cofunction relationship changes the angle.

An angle outside the acute range also illustrates the algebra: sin(120°) becomes cos(−30°). Both equal √3/2. The negative complementary angle does not by itself make the identity invalid.

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Why Use This Calculator?

Read the answer as an alternative way to write the input expression. The degree symbol identifies the unit. If a question asks for an exact numerical value, you still need to evaluate the resulting function using a known angle or another suitable method.

The identity does not create a value where the original function is undefined. For example, tan(90°) rewrites to cot(0°), and both are undefined. The current calculator can return the text cot(0°) without a warning. Check the domain before treating every displayed expression as a finite answer.

The sine/cosine pair is defined for all real angles. Tangent and secant require a nonzero cosine; cotangent and cosecant require a nonzero sine. These restrictions explain why a formally correct rewrite can still represent an undefined expression.

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Common Mistakes

    Do not confuse complementary angles, which add to 90 degrees, with supplementary angles, which add to 180 degrees. Subtracting from 180 belongs to different trig relationships and may change a sign. Also avoid keeping the function unchanged: sin(30°) is not generally equal to sin(60°).

    Keep the original angle unit consistent throughout the subtraction. Entering 1.5708 because a problem says π/2 radians makes the tool treat that number as degrees. Finally, retain signs when subtracting a negative input: 90 − (−20) = 110, not 70.

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Frequently Asked Questions

Is cofunction the same as reciprocal function?

No. A reciprocal relationship gives sec(x) = 1/cos(x) at the same angle. A cofunction relationship gives sec(x) = csc(90° − x) at a different angle. The matching names can make these two ideas easy to confuse.

Why does the result show a function instead of a decimal?

The purpose of this tool is to rewrite the expression. It calculates only the complementary angle and selects the matching function name. A result such as sin(37°) is already the requested symbolic answer, even though its decimal value is not displayed.

Can I use an angle of 45 degrees?

Yes. Its complement is also 45 degrees. Sine and cosine have equal values there, and tangent and cotangent both equal one. The function name can change even when the angle stays the same.

Does applying the operation twice return the original?

Yes, on the common domain. Subtracting twice gives 90 − (90 − x) = x, and each cofunction pair switches back to its starting member. This is a useful check on the function pairing and the subtraction.

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Sources & References

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