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.