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

giorgiana1976 | Student

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.

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