**Table of Contents for this page**

**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?

**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

tangent and cotangent

secant and cosecant

### 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} }}) } $

**Table of Contents for this page**