# Graphing Trigonometric Functions with Domain and Range

## Key Questions

Using a graphing calculator: MODE must be in radians

#### Explanation:

Using a graphing calculator: MODE must be in radians.

On a TI graphing calculator, with the standard zoom, $Y 1 = \sin \left(x\right)$:

graph{sin(x) [-10.04, 9.96, -5.16, 4.84]}

On a TI graphing calculator, with the standard zoom, $Y 1 = \cos \left(x\right)$:

graph{cosx [-10.04, 9.96, -5.16, 4.84]}

On a TI graphing calculator, with the standard zoom, $Y 1 = \tan \left(x\right)$:

graph{tan x [-10.04, 9.96, -5.16, 4.84]}

On a TI graphing calculator, with the standard zoom, $y = \sec \left(x\right) : Y 1 = \frac{1}{\cos \left(x\right)}$:

graph{1/(cos x) [-10.04, 9.96, -5.16, 4.84]}

On a TI graphing calculator, with the standard zoom, $y = \csc \left(x\right) : Y 1 = \frac{1}{\sin \left(x\right)}$:

graph{1/(sin x) [-10.04, 9.96, -5.16, 4.84]}

On a TI graphing calculator, with the standard zoom, $y = \cot \left(x\right) : Y 1 = \frac{1}{\tan \left(x\right)}$:

graph{1/(tan x) [-10.04, 9.96, -5.16, 4.84]}

$\sin \left(x\right)$, $\cos \left(x\right)$, $\tan \left(x\right)$, $\csc \left(x\right)$, $\sec \left(x\right)$, $\cot \left(x\right)$.

#### Explanation:

$\cos \left(x\right) = \sin \left(\frac{\pi}{2} - x\right)$

$\tan \left(x\right) = \frac{\sin \left(x\right)}{\cos \left(x\right)}$

$\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$

$\sec \left(x\right) = \frac{1}{\cos} \left(x\right)$

$\cot \left(x\right) = \frac{1}{\tan} \left(x\right)$

• Given a right triangle, select a vertex where the hypotenuse and one of the legs meet.

The angle $\theta$ created by this two sides will be used as a reference point

The side that formed the angle $\theta$ together with the hypotenuse will be referred to as $a \mathrm{dj} a c e n t$ (side adjacent to the angle). The other side will be referred to as $o p p o s i t e$ (side opposite the angle)

The ratio between the $o p p o s i t e$ and the $\text{hypotenuse}$ is called "sine" (sin). The inverse of this ratio is called "cosecant" (csc)

$\sin \theta = \text{opposite" / "hypotenuse}$

$\csc \theta = \text{hypotenuse" / "opposite} = \frac{1}{\sin} \theta$

The ratio between the $a \mathrm{dj} a c e n t$ and the $\text{hypotenuse}$ is called "cosine". The inverse of this ratio is called "secant"

$\cos \theta = \text{adjacent" / "hypotenuse}$

$\sec \theta = \text{hypotenuse" / "adjacent} = \frac{1}{\cos} \theta$

The ratio between the $o p p o s i t e$ and the $a \mathrm{dj} a c e n t$ is called
$\text{tangent}$. The inverse of this ratio is called $\text{cotangent}$

$\tan \theta = \text{opposite" / "adjacent}$

$\cot \theta = \text{adjacent" / "opposite} = \frac{1}{\tan} \theta$

For example, in a 30-60-90 triangle

$\sin 30 = \frac{1}{2}$
$\cos 30 = {3}^{\frac{1}{2}} / 2$
$\tan 30 = \frac{1}{3} ^ \left(\frac{1}{2}\right)$
$\csc 30 = 2$
$\sec 30 = \frac{2}{3} ^ \left(\frac{1}{2}\right)$
$\cot 30 = {3}^{\frac{1}{2}}$

$\sin 60 = {3}^{\frac{1}{2}} / 2$
$\cos 60 = \frac{1}{2}$
$\tan 60 = {3}^{\frac{1}{2}}$
$\csc 60 = \frac{2}{3} ^ \left(\frac{1}{2}\right)$
$\sec 60 = 2$
$\cot 60 = \frac{1}{3} ^ \left(\frac{1}{2}\right)$