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

The general solution for cos(x)+cos^4(x)= 1/2 is x=1.09667…+2pin,x=2pi-1.09667…+2pin

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

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

x = 1.09667 \dots + 2 πn, x = 2 π - 1.09667 \dots + 2 πn

Solution steps

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

Solve by substitution

cos (x) \approx 0.45655 \dots, cos (x) \approx - 1.12990 \dots

cos (x) = 0.45655 \dots : x = arccos (0.45655 \dots) + 2 πn, x = 2 π - arccos (0.45655 \dots) + 2 πn

cos (x) = - 1.12990 \dots :

No Solution

Combine all the solutions

x = arccos (0.45655 \dots) + 2 πn, x = 2 π - arccos (0.45655 \dots) + 2 πn

Show solutions in decimal form

x = 1.09667 \dots + 2 πn, x = 2 π - 1.09667 \dots + 2 πn

sin^{2} (x) = sec (x)

5 sin^{2} (x) - 11 sin (x) + 2 = 0

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

3 sin (a) + cos (a) = 1

4 cos^{2} (x) + 3 = 2 (3 + 1)

What is the general solution for cos(x)+cos^4(x)= 1/2 ?
The general solution for cos(x)+cos^4(x)= 1/2 is x=1.09667…+2pin,x=2pi-1.09667…+2pin