trigonometric functions

In mathematics, the trigonometric functions are functions which relate an angle of a right-angled triangle to ratios of two side lengths.

The unit circle is used to visualize the outputs of the trig functions

In case you are wondering why trig functions are important, we use them a lot in geometric problems and calculus.

introduction

some easy applications of basic trigonometry is in introduction to trigonometry

figure: we should be familiar with triangle stuff

We probably know soh cah toa already

sine
$\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}$

cosine
$\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}$

tangent
$\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}$
$\tan (\theta )=\frac{\sin (\theta )}{\cos (\theta )}$

how to solve equations with trig functions

how to evaluate trig functions
how to graph trig functions

notation

trig functions can leave out the brackets where appropriate, like depicting $\sin (x)$ as $\sin x$

a superscript right after the name of a trig function denotes exponentiation, for example, ${\sin }^2(x)=(\sin (x){)}^2$
this is different to function composition where $f^2(x)=(f\circ f)(x)=f(f(x))$

reciprocal trig functions

sine, cosine and tangent trig functions have their reciprocals

related

• how to evaluate reciprocal trig functions
• how to graph reciprocal trig functions

cosecant
$\csc (\theta )=\frac{1}{\sin (\theta )}$

secant
$\sec (\theta )=\frac{1}{\cos (\theta )}$

cotangent
$\cot (\theta )=\frac{1}{\tan (\theta )}$
$\cot (\theta )=\frac{\cos (\theta )}{\sin (\theta )}$

inverse trig functions

related

• how to evaluate inverse trig functions
• how to graph inverse trig functions

the inverse trig functions are not the same as trig functions

whereas in trig function you input an angle and get a ratio, with inverse trig functions you input a ratio and get an angle

since the trig functions are not one-to-one (see relation), the inverse trig functions have to restrict their domain

inverse sine is only a segment of the sine function, the $[-\frac{\pi }{2},\frac{\pi }{2}]$ segment, and it’s the inverse function of this segment
remember the range and how the output of inverse tan always has the same sign as the input

$\sin (x)$
domain: $\mathbb{R}$
range: $[-1,1]$

${\sin }^{-1}(x)$
domain: $[-1,1]$
range: $[-\frac{\pi }{2},\frac{\pi }{2}]$

the inverse cosine is similar
remember the range and how the output of inverse cosine is always non-negative

$\cos (x)$
domain: $\mathbb{R}$
range: $[-1,1]$

${\cos }^{-1}(x)$
domain: $[-1,1]$
range: $[0,\pi ]$

the inverse tan has the same range as inverse sine
remember the range and how the output of inverse tan always has the same sign as the input

$\tan (x)$
domain: $\mathbb{R}$
range: $\mathbb{R}$

${\tan }^{-1}(x)$
domain: $\mathbb{R}$
range: $[-\frac{\pi }{2},\frac{\pi }{2}]$