how to evaluate reciprocal trig functions

in how to evaluate trig functions, we used stuff to evaluate

we know that besides the sine and cosine, we have the other trig functions that are so useless and we have to convert them into sin and cosine all the time

below is the definitions of all the trig functions

sine
$\sin (\theta )=\frac{o}{h}$

cosine
$\cos (\theta )=\frac{a}{h}$

tangent
$\tan (\theta )=\frac{o}{a}$

cosecant
$\csc (\theta )=\frac{h}{o}$

secant
$\sec (\theta )=\frac{h}{a}$

cotangent
$\cot (\theta )=\frac{a}{o}$

as you can see, everything can be expressed as sine and cosine functions

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

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

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

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

how do we remember the names of these other trig functions?

how i remember it is to say the name of the trig function out loud, and the third letter of the trig function name is either s, c or t, which are inverses

like, csc is cosecant, which is the inverse of sine, $\csc (a)=\frac{1}{\sin (a)}$
sec is secant, which is the inverse of cosine, $\sec (a)=\frac{1}{\cos (a)}$

evaluating these other trig functions isn’t hard, for you also find the absolute value of the output, and then you apply the correct sign

we can also reference the common angles table

$\theta$	$0$	$\frac{\pi }{6}$	$\frac{\pi }{4}$	$\frac{\pi }{3}$	$\frac{\pi }{2}$
$\sin (\theta )$	0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1
$\cos (\theta )$	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0
$\tan (\theta )$	0	$\frac{1}{\sqrt{3}}$	1	$\sqrt{3}$	undefined

worked example: evaluate $\csc (\frac{3\pi }{4})$
cosecant corresponds to sine
applying identities
$\frac{3\pi }{4}$ is in the second quadrant, which has the same sine value as $\frac{\pi }{4}$
$\csc (\frac{\pi }{4})=\frac{1}{\sin (\frac{\pi }{4})}$

$\csc (\frac{3\pi }{4})$

looking at the common angles table, $\frac{3\pi }{4}$ matches with the $\frac{\pi }{4}$ column

$\sin (\frac{\pi }{4})=\frac{\sqrt{2}}{2}$

so $|\csc (\frac{3\pi }{4})|=\sqrt{2}$

looking at the signs for the quadrants, $\frac{3\pi }{4}$ is in the second quadrant, where only sine is positive, and the reciprocal of sine is also positive
$\csc (\frac{3\pi }{4})=\sqrt{2}$

worked example: evaluate $\sec (\frac{5\pi }{6})$
secant corresponds to cosine
applying identities
$\frac{5\pi }{6}$ is in the second quadrant, having the same aboslute

looking at the common angles table, $\frac{5\pi }{6}$ matches with the $\frac{\pi }{6}$ column
$\cos (\frac{\pi }{6})=\frac{\sqrt{3}}{2}$
so

worked example: evaluate $\cot (-\frac{\pi }{3})$
cotagent corresponds to tangent
looking at the common angles table at the $\frac{\pi }{3}$ column
$\tan (\frac{\pi }{3})=\sqrt{3}$