# 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(0°) = 1

### Sine Cosine on a TI Calculator

#### 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?

• 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.

#### So, what does that mean?

• 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

### Concept Questions

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:

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

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:

### Practice Problems

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

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.

True .
Yes this solves the two requirements of cofunctions

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

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

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

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 $-cos(30^{\circ}) = - sin(60^{\circ} ) \\ \boxed{ - \sin( {\red{60^{\circ} }}) }$

$$265^{\circ}$$ 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 $tan( \red{ 85^{\circ}}) = cot( \red{ 90-85^{\circ } }) \\ \boxed{ \cot( {\red{5^{\circ} }}) }$