If sinx + 3^1/2*cosx = 1 . What is x ?

This is linear equation and we'll solve it using the
helping angle method.

sinx + sqrt3*cosx =
1

We notice that the coefficients of cos x, namely sqrt 3,
could be written as tan pi/3.

We'll substitute sqrt 3 by
tan pi/3 and we'll get:

sinx + tan pi/3*cosx =
1

We'll write tan pi/3 = sin pi/3/cos
pi/3

We'll substitute tan pi/3 by the ratio sin pi/3/cos
pi/3.

sinx + (sin pi/3/cos pi/3)*cosx =
1

We'll multiply by cos pi/3 the terms without
denominator:

sin x*cos pi/3 + sin pi/3*cosx = cos
pi/3

The sum from the left side is the result of applying
the formula;

sin (a+b) = sina*cosb + sinb*cos
a

We'll put a = x and b =
pi/3

sin x*cos pi/3 + sin pi/3*cosx = sin (x +
pi/3)

We'll substitute the sum by the sine
function:

sin (x + pi/3) = cos
pi/3

sin (x + pi/3) = 1/2

x +
pi/3 = (-1)^k*arcsin (1/2) + k*pi

x + pi/3 =
pi/6

x = pi/3 - pi/6

x = (2pi
- pi)/6

x =
pi/6

The solution of the
equation is x = pi/6.