√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 = 12).

How to find the general solution of the trigonometric equation cos x = 12 or √2 cos x - 1 = 0?

Solution:

We have,

√2 cos x - 1 = 0

⇒ √2 cos x = 1

⇒ cos x = 12

⇒ cos x = cos π4  or, cos (- π4)

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

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, π2, π, 3π2, 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 = 12 

⇒ cos x = cos π4

⇒ cos x = cos (2nπ + π4), Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Therefore, x = cos (2nπ + π4) …………….. (i)

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

cos x = 12  

⇒ cos x = cos (- π4

⇒ cos x = cos (2nπ - π4), Where n ∈ I (i.e., n = 0, ± 1, ± 2, ± 3,…….)

Therefore, x = cos (2nπ + π4) …………….. (ii)

Therefore, the general solutions of equation cos x = 12 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 ± π4, n ∈ I.

 Trigonometric Equations






11 and 12 Grade Math

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