√2 cos x - 1 = 0

We will discuss about the general solution of the equation square root of 2 cos x minus 1 equals 0 (i.e., √2 cos x - 1 = 0) or cos x equals 1 by square root of 2 (i.e., cos x = $\frac{1}{√2}$).

How to find the general solution of the trigonometric equation cos x = $\frac{1}{√2}$ or √2 cos x - 1 = 0?

Solution:

We have,

√2 cos x - 1 = 0

⇒ √2 cos x = 1

⇒ cos x = $\frac{1}{√2}$

⇒ cos x = cos $\frac{π}{4}$  or, cos (- $\frac{π}{4}$)

Let O be the centre of a unit circle. We know that in unit circle, the length of the circumference is 2π.

√2 cos x - 1 = 0

If we started from A and moves in anticlockwise direction then at the points A, B, A', B' and A, the arc length travelled are 0, $\frac{π}{2}$, π, $\frac{3π}{2}$, and 2π.

Therefore, from the above unit circle it is clear that the final arm OP of the angle x lies either in the first or in the fourth quadrant.

If the final arm OP lies in the first quadrant then, 

cos x = $\frac{1}{√2}$ 

⇒ cos x = cos $\frac{π}{4}$

⇒ cos x = cos (2nπ + $\frac{π}{4}$), Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Therefore, x = cos (2nπ + $\frac{π}{4}$) …………….. (i)

Again, if the final arm OP of the unit circle lies in the fourth quadrant then,

cos x = $\frac{1}{√2}$  

⇒ cos x = cos (- $\frac{π}{4}$) 

⇒ cos x = cos (2nπ - $\frac{π}{4}$), Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Therefore, x = cos (2nπ + $\frac{π}{4}$) …………….. (ii)

Therefore, the general solutions of equation cos x = $\frac{1}{√2}$ are the infinite sets of value of x given in (i) and (ii).

Hence general solution of √2 cos x - 1 = 0 is x = 2nπ ± $\frac{π}{4}$, n ∈ I.

● Trigonometric Equations

• General solution of the equation sin x = ½
• General solution of the equation cos x = 1/√2
• General solution of the equation tan x = √3
• General Solution of the Equation sin θ = 0
• General Solution of the Equation cos θ = 0
• General Solution of the Equation tan θ = 0
• General Solution of the Equation sin θ = sin ∝
  
• General Solution of the Equation sin θ = 1
• General Solution of the Equation sin θ = -1
• General Solution of the Equation cos θ = cos ∝
• General Solution of the Equation cos θ = 1
• General Solution of the Equation cos θ = -1
• General Solution of the Equation tan θ = tan ∝
• General Solution of a cos θ + b sin θ = c
• Trigonometric Equation Formula
• Trigonometric Equation using Formula
• General solution of Trigonometric Equation
• Problems on Trigonometric Equation

11 and 12 Grade Math

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