Cofunctions: Definition & Examples

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Cofunctions in trigonometry are function pairs like sine and cosine. In this lesson we define the common trig cofunctions and use examples to show how they are used.

Cofunctions

You and the world-famous but imaginary Cosinius Matham have decided to coauthor a thrilling high-adventure screenplay featuring trigonometry. This will not be your basic math movie and your writing will be complemented by Cosinius. For whatever plot angle you suggest, Cosinius will write a complementary angle.

In this lesson, we look at trig functions and complementary angles. I wonder who will costar in this epic film.

Cosine and Sine

The words 'coauthor' and 'costar' start with the prefix 'co'. So do the words 'cofunction' and 'complementary.' In math, two angles are complementary if they add to 90o. For example, 40o and 50o are complementary.

We have three fundamental trig functions: sine, cosine and tangent. There's a 'co' at the start of the word 'cosine' suggesting something special about cosine and sine: they are cofunctions.

Remember those complementary angles 40o and 50o? Cosine of 40o is equal to sine of 50o. Check it out with your calculator. Cos(40o) = .766 and sin(50o) = .766.

What about sine of 31o? Let's see, 31o plus what angle is equal to 90o? Another way to think about this is 90 - 31 is 59. Aha! sin(31o) should equal cos(59o). Time for a calculator check: sin(31o) = .515 and cos(59o) = .515. It checks again!

Instead of a specific angle, let's use θ. If one of the angles is θ, the complementary angle is 90o - θ. This is cool because θ plus the other angle, 90o - θ, add together to equal θ + (90o - θ) = 90o . Time to write two equations:

1. sin(θ) = cos(90o - θ)

2. cos(θ) = sin(90o - θ)

The first equation makes sense looking at a right triangle.


Verifying equation 1
right-triangle


A triangle has three angles. The sum of the three angles is 180o. In a right-triangle, one of the angles is 90o leaving us with another 90o to account for. If one of those angles is θ, the other angle must be 90o minus θ.

Remembering the acronym 'SohCahToa', sine is the opposite side over the hypotenuse (the 'Soh') while cosine is the adjacent over the hypotenuse (the 'Cah'). In the figure, the side opposite the angle θ is a and the hypotenuse is c meaning sin(θ) = opposite/hypotenuse = a/c. What about the angle 90o - θ? The side adjacent to the angle 90o - θ is a meaning cos(90o - θ) = adjacent/hypotenuse = a/c. We get a/c in both cases.

With a right-triangle and the definitions of sine and cosine we have shown sin(θ) = cos(90o - θ). Similarly, the second equation can be verified by showing both cos(θ) and sin(90o - θ) are equal to b/c.

Example 1

If cos(49o) = sin(θ), what is θ?

Thinking this through: cosine and sine are cofunctions; the 'co' prefix reminds us of complementary. We are looking for two angles adding to 90o. Since 49o is one of the angles, the other angle, θ, is 90o - 49o = 41o. Then, cos(49o) = sin(41o). Verifying this with a calculator,

cos(49o) = 0.656

sin(41o) = 0.656

Cosinius Mathma needs more characters for the movie. Any other cofunctions?

Cotangent and Tangent

As you might have guessed, cotangent and tangent are cofunctions. Giving us:

3. tan(θ) = cot(90o - θ)

4. cot(θ) = tan(90o - θ)

Some facts about tangent and cotangent:

  • tangent = sine/cosine
  • cotangent = 1/tangent = cosine/sine

Tan(θ) is sin(θ)/cos(θ). Using the cofunction equations for sine and cosine: sin(θ)/cos(θ) = cos(90o - θ)/sin(90o - θ).

But cosine/sine is cotangent giving cos(90o - θ)/sin(90o - θ) = cot(90o - θ). Thus, tan(θ) = cot(90o - θ).

Example 2

cot(θ) = tan(105o). What is θ?

Tangent and cotangent are cofunctions meaning θ plus 105o equals 90o. Thus, θ = -15o.

Check:

tan(105o) = -3.732

cot(-15o) = -3.732

If your calculator does not have a cotangent button, you can compute cot(-15o) as 1/tan(-15o).

Anymore trig cofunctions? What about cosecant and secant? I wonder what the role will be for cosecant and secant.

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