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

The general solution for sin^2(x)-cos(x)= 1/2 is x=1.19606…+2pin,x=2pi-1.19606…+2pin

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

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

x = 1.19606 \dots + 2 πn, x = 2 π - 1.19606 \dots + 2 πn

Solution steps

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

Subtract

\frac{1}{2}

from both sides

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

Simplify

sin^{2} (x) - cos (x) - \frac{1}{2} : \frac{2 sin ^{2} ( x ) - 2 cos ( x ) - 1}{2}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

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

Solve by substitution

cos (x) = - \frac{1 + 3}{2}, cos (x) = \frac{3 - 1}{2}

cos (x) = - \frac{1 + 3}{2} :

No Solution

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

Combine all the solutions

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

Show solutions in decimal form

x = 1.19606 \dots + 2 πn, x = 2 π - 1.19606 \dots + 2 πn

cos (x) [3 sin (x) - 2] = 0

sin^{3} (x) + sin (x) - 4 = 0

a, sin^{2} (a) + cos^{2} (b) = 1

sin^{4} (x) = - cos (x)

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

What is the general solution for sin^2(x)-cos(x)= 1/2 ?
The general solution for sin^2(x)-cos(x)= 1/2 is x=1.19606…+2pin,x=2pi-1.19606…+2pin