What is a cofunction?

Before we tackle what a cofunction is, you should be comfortable with definition of a function and what complementary angles are.

Explore the pattern for

Cofunctions

Can you see a pattern below? Pay attention to the values of sine and cosine and their angle measurements.
$$ \theta $$ sin( $$ \theta $$ ) $$ \theta $$ - 90 cos($$ \theta $$- 90 )
10 ° sin(10 ) = 0.17364817766693 90 - 10 = 80 ° cos(80°) = 0.17364817766693
20 ° sin(20 ) = 0.34202014332567 90 - 20 = 70 ° cos(70°) = 0.34202014332567
30 ° sin(30 ) = 0.5 90 - 30 = 60 ° cos(60°) = 0.5
40 ° sin(40 ) = 0.64278760968654 90 - 40 = 50 ° cos(50°) = 0.64278760968654
50 ° sin(50 ) = 0.76604444311898 90 - 50 = 40 ° cos(40°) = 0.76604444311898
60 ° sin(60 ) = 0.86602540378444 90 - 60 = 30 ° cos(30°) = 0.86602540378444
70 ° sin(70 ) = 0.93969262078591 90 - 70 = 20 ° cos(20°) = 0.93969262078591
80 ° sin(80 ) = 0.98480775301221 90 - 80 = 10 ° cos(10°) = 0.98480775301221
90 ° sin(90 ) = 1 90 - 90 = 0 ° cos() = 1

Sine Cosine

on a TI Calculator

ti calculator degree mode, then sine cosine cofunction

So, did you notice the pattern?

Every output value of sin $$ \theta$$ is exactly equal to the output value of cos(90 - $$\theta$$)

So, what is a cofunction?

diagram of cofunction
  • Two functions whose complementary input angles $$ \rightarrow $$ evaluate to equal output
  • It's important to note that the functions are not complementary
  • Complementary input produces equal output.
    pair of cofunctions input output example
    pair of cofunctions input output example

So, what does that mean?

example of complementary angles
  • In summary: Input is complementary and output is equal.

Most Common

Cofunction Formulas

sine and cosine

Degree example $$ sin ( \theta ) = cos ( \red {90}- \theta ) $$ $$ cos( \theta ) = sin( \red {90} - \theta ) $$
Radian example $$ sin ( \theta ) = cos ( \red { \frac{\pi}{2}}- \theta ) $$ $$ cos( \theta ) = sin( \red {\frac{\pi}{2}} - \theta ) $$

tangent and cotangent


Degree example $$ tan( \theta ) = cot( \red {90}- \theta ) $$ $$ cot( \theta ) = tan( \red {90} - \theta ) $$
Radian example $$ tan( \theta ) = cot ( \red { \frac{\pi}{2}}- \theta ) $$ $$ cot( \theta ) = tan( \red {\frac{\pi}{2}} - \theta ) $$

secant and cosecant


Degree example $$ sec( \theta ) = csc( \red {90}- \theta ) $$ $$ csc( \theta ) = sec( \red {90} - \theta ) $$
Radian example $$ sec( \theta ) = csc ( \red { \frac{\pi}{2}}- \theta ) $$ $$ csc( \theta ) = sec( \red {\frac{\pi}{2}} - \theta ) $$

Cofunctions Graphically

You can see the relationship between sine and cosine, graphically, when you plot $$ sin( \theta) $$ and $$ cos( \theta)$$ on the same set of axes
 graph of sine and cosine as cofuncitons

Concept Questions

Problem 1

Is the statement below True or False

Statement: Since sine and cosine are cofunctions, they are complementary

Answer

wrong False.
The functions themselves (sine and cosine) are not complementary . Being a cofunction, means that complementary input angles leads to the same output , as shown in the following example:
complementary  input

Problem 2

Is the statement below True or False?

Statement: $$ f(x) \text{ and } g(x) $$ are cofunctions because $$ g(60^{\circ}) = .5 $$ and $$ f( 30^{\circ}) = .5 $$

Answer

wrongTrue.
Since the input angles are complementary and yield the same output, $$f(x)$$ and $$ g(x) $$ are cofunctions.
complementary  input

In this case, we can even figure out what the functions for $$ g(x)$$ and $$ f(x)$$ are :


$ g = sin \\ f= cos $ You can try for yourself, by entering the following values into your calculator: $ g( 60^{\circ}) = cos (60^{\circ} ) = .5 \\ f( 30^{\circ}) = sin (30^{\circ} ) = .5 $

Still need more convincing, here's a reminder of the animated gif showing a TI graphing calculator showing these very calculations:
ti calculator degree mode, then sine cosine cofunction

Practice Problems

Problem 1

Find a value of $$ \theta $$ for which the equation below is true

Equation: $$ sin ( \theta ) = cos( 15^{\circ}) $$

Answer

Since sine and cosine are cofunction, we know that the angles are complementary ; $ sin ( \red \theta ) = cos( \red{ 15^{\circ} } ) \\ \red \theta + \red{ 15^{\circ}} = 90 ^{\circ} \\ \red \theta = 90 ^{\circ} - 15^{\circ } \\ \theta = \boxed { 75^{\circ }} $

Problem 2

Is the statement below True or False?

Statement: $$ f(x) $$ and $$ g(x) $$ are cofunctions because $$ f( \theta) = \red{20} $$ and $$ g( \theta) = \red{60} $$.

Answer

wrong False.
The output of cofunctions is not complementary!!! We need complementary input to give us ( or evaluate to if you want to use technical math speak) equal output.

Problem 3

Is the statement below True or False

Statement: Tangent and cotangent are cofunctions because $$ tan( \red{\theta} ) = 1.2 $$ and $$ cot( \red{ 90 - \theta} ) = 1.2 $$

Answer

wrong True .
Yes this solves the two requirements of cofunctions

  • complementary input $$ \rightarrow$$ yields equal output

Problem 4

Write the expression $$ cos(80) $$ as the function of an acute angle of measure less than $$45^{\circ} $$ .

Answer

Thanks, to our cofunction formula, we know that $$ cos(80^{\circ}) = sin(10^{\circ}) $$

Problem 5

Write the expression $$ cos(210) $$ as the function of an acute angle, measuring greater than $$45^{\circ} $$ .

Answer

$$ 210^{\circ} $$ is in the third quadrant with a reference angle of $$ 30^{\circ} $$ .
reference angle for 210 degrees


So, we know that $$ cos(210^{\circ}) = \red - cos(30^{\circ}) $$ To get an angle greater than $$ 45^{\circ}$$, we can use the cofunction of cosine: sine $ -cos(30^{\circ}) = - sin(60^{\circ} ) \\ \boxed{ - \sin( {\red{60^{\circ} }}) } $

Problem 6

Write the expression $$ sec(265^{\circ}) $$ as the function of an acute angle of measure less than $$45^{\circ} $$ .

Answer

$$ 265^{\circ} $$ is in the third quadrant with a reference angle of $$ 85^{\circ} $$ .


So, we know that $$ tan(265^{\circ}) = tan(85^{\circ}) $$ To get an angle less than $$ 45^{\circ}$$, we can use the cofunction of tangent : cotangent $ tan( \red{ 85^{\circ}}) = cot( \red{ 90-85^{\circ } }) \\ \boxed{ \cot( {\red{5^{\circ} }}) } $

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