Question

How do you evaluate if possible the 6 trigonometric functions of the real number `t= (7pi)/6`?

Answer 1

`sin ((7pi)/.6))=-1/2, cos ((7pi)/6)= -sqrt3/2 and tan ((7pi)/6)=1/sqrt3`.
The recprocals, `csc ((7pi)/.6))=-2, sec ((7pi)/6)= -2/sqrt3 and cot ((7pi)/6)=sqrt3`...

Explanation:

In the third quadrant, both sine and cosine are negative and the tangent is positive.

`7pi/6` direction is in the third quadrant.

For any of the six functions `f(theta)`, `f(pi+theta)=+-f(theta) and 7pi/6=pi+pi/6`.

So, in every case it is `+-f(pi/6)`.
Choose appropriate sign and prefix to `f(pi/6)`. ..