What is the general solution for 4sin^2(x)=cos(x)+1 ?

The general solution for 4sin^2(x)=cos(x)+1 is x=pi+2pin,x=0.72273…+2pin,x=2pi-0.72273…+2pin

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4sin^2(x)=cos(x)+1

4 sin^{2} (x) = cos (x) + 1

x = π + 2 πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

Solution steps

4 sin^{2} (x) = cos (x) + 1

Subtract

cos (x) + 1

from both sides

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

Rewrite using trig identities

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

Solve by substitution

cos (x) = - 1, cos (x) = \frac{3}{4}

cos (x) = - 1 : x = π + 2 πn

cos (x) = \frac{3}{4} : x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

Combine all the solutions

x = π + 2 πn, x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

Show solutions in decimal form

x = π + 2 πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

0 = 4 sin (x)

tan (- 6 0^{\circ}) = \frac{( 1 - cos ( u ) )}{sin ( u )}

sec^{2} (θ) (1 + cos^{2} (θ)) = 2 - tan^{2} (θ)

sin (2 x) sin (x) - cos (2 x) cos (x) = - cos (x)

8 tan (x) - 12 = 5 tan (x) - 9

What is the general solution for 4sin^2(x)=cos(x)+1 ?
The general solution for 4sin^2(x)=cos(x)+1 is x=pi+2pin,x=0.72273…+2pin,x=2pi-0.72273…+2pin