How to find the values of trigonometrical functions for any angle in radians In order to find the values of trigonometrical functions for any angle in terms of those of positive acute angle, we have the following algorithm (from my textbook):

ALGORITHM
STEP I: See whether the given angle a is positive or negative if it is negative, make it positive by using the following:
$$\sin(-θ) = -\sin θ,\ \cos(-θ) = \cos θ,\ \tan(-θ)= -\tan θ,\ \text{etc.}$$
STEP II: Express the positive angle a obtained in step I in the form $α=90°n ± θ$, where $θ$ is an acute angle.
STEP III: Determine the quadrant in which the terminal side of the angle $α$ lies.
STEP IV: Determine the sign of the given trigonometrical function in the quadrant obtained in step III.
STEP V: If $n$ in step II is an odd integer, then
\begin{gather*}
\sin α =±\cos θ,\ \cos α =±\sin θ,\ \tan α =±\cot θ,\\
\sec α = ±\csc θ,\ \csc α = ±\sec θ.
\end{gather*}
And the sign on RHS will be the sign obtained in step IV. If $n$ in step II is an even integer, then
\begin{gather*}
\sin α=±\sin θ,\ \cos α =±\cos θ,\ \tan α = ±\tan θ,\\
\sec α= ±\sec θ, \ \csc α = ±\csc θ.
\end{gather*}
And the sign on RHS is the sign obtained in step IV.

I am not able to proceed through step II & III, V. If the angle is given in degrees, then I can apply Euclid's division algorithm to express $α=90°n ± θ$, e.g. $590°= 90° × 6 + 30°$. But I am having difficulty if the angle is given in radians (for example $\dfrac{11π}{8}$). Of course I can convert it to degrees but I don't want to.
In step III, how to determine the quadrant in which an angle lies? I am not able to determine the quadrant especially if the angle is very large. Once I know the quadrant I can easily determine the sign by using All Students Take Calculus mnemonic.
So how to find the value of any trigonometric function at any angle in radians and/or degrees? And if you know any tricks for evaluating trigonometrical values then please share it (with an example). Thank you.
 A: It is actually quite simple
Based on your question,I think you have a problem regarding two things

*

*You are able to simplify if the angles are in degrees but you are getting stuck in which quadrant should I place the angle

2)Of the angles are in radians you are not able to solve them
So,
For problem 1)
Suppose an angle like 570°
What you did was correct applying Euclid division algorithm
So it would be $570°=90*6+ 30°$
That implies angle(radians) is $(\frac{6\pi}{2} + 30°)$
Upon simplification,$(3\pi +30°)$
What I used to do to remember the quadrants
If the given radian is like $(n\pi)$
Then if $n$ is even then the it is in $+^{ve} $x direction
If $n$ is odd then it is in $-^{ve}$ x direction
Further explanation on how to solve:
If the given angle is like $(n\pi + 30°)$ where 'n' is odd
Then the angle would be $3^{rd}$ quadrant because there is an increase of 30° from $-^{ve} $ x-direction
If it is $(n\pi -30°)$ where n is odd then the angle would lie in $2^{nd}$ quadrant
So similarly if angle is like $\frac{n\pi}{2}$ where n is odd then it there will be two cases
1)Where n is in the form $(4x+1)$ Then it lies in $+^{ve} $ y direction $(x\geq 0)$
For example:$[4(0)+1][\frac{\pi}{2}]$
Then it is $\frac{\pi}{2}$
2)Where n is of the form $(4x-1)$ Then it lies in $-^{ve}$ y direction where $(x\geq 1)$
For example :$(\frac{3\pi}{2})$
Use the same principle as stated for the $n\pi$  logic
This will be enough to solve all the questions related to this
If my answer is not clear,
Then tell me where you need clarity.
For your $2^{nd}$ problem it is pretty simple
In you question you have given $\frac{11\pi}{8}$
The main motive to simplify is to solve trigonometric functions right
So we can apply half angle formula for the given function
Suppose the function is
$ \sin(\frac{11\pi}{8}$
Then we can just simplify through half angle formula that is
$\sin(A) = 2\sin(\frac{A}{2})\cos(\frac{A}{2})$
But if the angle is given as $\frac{613\pi}{2}+\theta$
Then divide 613 by 4 and the
Remainder will determine how to simplify
Two cases:
1)If remainder is 1 then it lies in $-^{ve}$ y direction
2)If remainder is 3 then it lies in $+^{ve} $ y direction
Then I think you can solve through previous mentioned methods but anyway I will solve
$\frac{613\pi}{2}+\theta$
First thing divide 613 by 4
Remainder you will get is 1
So the angle lies in $-^{ve}$ y direction
As it lies in $-^{ve}$ y direction  adding some angle $\theta$ will result in the function being in $4^{th}$ quadrant
If the question is $\sin(\frac{613\pi}{2}+\theta)$
Then as $\frac{613\pi}{2}+\theta)$ lies in $4^{th}$ quadrant
The whole function will change into $-\cos(\theta)$
