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Popular >

cos(2x)2cos(x)=2(2cos^2(x)-1)2cos(x)

Solution

cos (2 x) 2 cos (x) = 2 (2 cos^{2} (x) - 1) 2 cos (x)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

+1

Solution steps

cos (2 x) \cdot 2 cos (x) = 2 (2 cos^{2} (x) - 1) \cdot 2 cos (x)

Subtract

2 (2 cos^{2} (x) - 1) 2 cos (x)

from both sides

2 cos (2 x) cos (x) - 4 cos (x) (2 cos^{2} (x) - 1) = 0

Factor

2 cos (2 x) cos (x) - 4 cos (x) (2 cos^{2} (x) - 1) : 2 cos (x) (cos (2 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1))

2 cos (x) (cos (2 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1)) = 0

Solving each part separately

cos (x) = 0 or cos (2 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (2 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1) = 0 : x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn, x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arccos (\frac{1}{2}) + 2 πn, x = 2 π - arccos (\frac{1}{2}) + 2 πn, x = arccos (- \frac{1}{2}) + 2 πn, x = - arccos (- \frac{1}{2}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.78539 \dots + 2 πn, x = 2 π - 0.78539 \dots + 2 πn, x = 2.35619 \dots + 2 πn, x = - 2.35619 \dots + 2 πn

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