Calculus How To

Cofunction: Definition

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Types of Functions >

Confunctions are complementary functions. They are usually used in trigonometry, and can be applied to the trigonometric functions. Sine and cosine (notice the “co” in that name?) are cofunctions, as are tangent and cotangent, and secant and cosecant.

A more formal definition:
If a function f is a cofunction of g, then the output of function g when evaluated at a particular angle will be equal to the output of f when evaluated at the angle complementary to that (complementary angles are two angles whose sum is 90°).

So if f is a cofunction of g, f(A) = g(B) whenever A and B are complementary angles.

Examples of Cofunction Relationships

You can see the cofunction identities in action if you plug a few values for sine and cosine into your calculator. The sine of ten° is 0.17364817766683; and this is exactly the same as the cosine of 80°. Note that 10 + 80 = 90; they are complementary angles.


The sine of 20° is 0.34202014332567; which is the same as the cosine of 70°.
The sine of 30° is 0.5, as is cosine of 60°.


You can see this graphically if you graph the two cofunctions, sine and cosine, on the same graph. They look like the same function, just shifted along the x axis.

cofunction relationships between sine and cosine

Unpacking Complementary Angles

Complementary angles are angle pairs whose sum adds up to a right angle: 90°, or π/2 radians. That makes the complementary angle to a 20 degree angle:
90 – 20 = 70 (because 70 + 20 = 90).
Similarly, the complementary angle of a π/6 angle is π/3 = (2π)/6, because (2π)/6 + π/6 = (3π)/6 = π/2.

The Cofunction Identities

The cofunction identities simply tell us the relationships between sine, tangent, and secant and their cofunctions:

References

Sum and Difference Formulas. Retrieved from https://www.alamo.edu/contentassets/35e1aad11a064ee2ae161ba2ae3b2559/analytic/math2412-sum-difference-indentities.pdf on January 27, 2018.
Summary of Trigonometric Identities. Retrieved from https://www2.clarku.edu/faculty/djoyce/trig/identities.html on January 27, 2018.

CITE THIS AS:
Stephanie Glen. "Cofunction: Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/cofunction/
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