You must go with series expansion(POWER Series) of those standard trigonometric functions.

If you consider TAN x expansion,

tan x = 1*x + 1/3*x^3 + 2/15*x^5 + ……….

So we must first convert our angle from degree to radians i.e. by dividing by 180° and MULTIPLYING by π that gives approximately 0.698 as the answer. Then by SUBSTITUTING 0.698 in the series expansion we can find tan 40° value(I have ignored the values of series tan x because our x here is less than 1 and as it gets raised to higher powers the value becomes smaller and smaller which we can neglect).

tan(.698) = 1*(0.698) + 1/3*(0.698)^3 + 2/15*(0.698)^5 + ……….

=> 0.698 + 0.1134 + 0.0221 ….

=> 0.8335

This is an approximate value and if we do it by calculator we get tan(40°) as 0.839 which is approximately equal to our answer itself.

So this answer applies to a general question of finding trigonometric angles of all standard as well as non-standard angles.

But for some cases we can find values just by doing arithmetic of standard known angles which uses trigonometric formulae.

Eg:- 15° = 45° - 30°, where both 45° and 30° of all trigonometric functions are known angles and widely used in problems also. So if asked to find tan 15° = tan(45° - 30°), takes the form of tan(A-B) which we know.

Series expansion of many functions :- Series Expansion