Find the value of sin β + 3cos β, given that sec β=5,and tan β=2√6. An exact value required?

Trigonometry is based on a picture of a circle with the radius making an angle with the x axis. You carry this picture in your head, not your calculator. Trig is mostly imagination and the rest memorization.

Remember Chief SohCahToa
Sin = Opposite/Hypotenuse
Cosine = Adjacent/Hypotenuse
Tangent = Opposite/Adjacent

Also memorize:
Sec = 1/Cos
Csc = 1/Sin
Cot = 1/Tan

You can use these memory aids to figure any trig function from given values, or just look them up in a table at en.wikipedia.org/wiki/List_of_trigonometric_identities