What is the general solution for sin(x)+cos^2(x)= 5/4 ?

The general solution for sin(x)+cos^2(x)= 5/4 is x= pi/6+2pin,x=(5pi)/6+2pin

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sin(x)+cos^2(x)= 5/4

sin (x) + cos^{2} (x) = \frac{5}{4}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

sin (x) + cos^{2} (x) = \frac{5}{4}

Subtract

\frac{5}{4}

from both sides

sin (x) + cos^{2} (x) - \frac{5}{4} = 0

Simplify

sin (x) + cos^{2} (x) - \frac{5}{4} : \frac{4 sin ( x ) + 4 cos ^{2} ( x ) - 5}{4}

\frac{4 sin ( x ) + 4 cos ^{2} ( x ) - 5}{4} = 0

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

4 sin (x) + 4 cos^{2} (x) - 5 = 0

Rewrite using trig identities

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

Solve by substitution

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 cos^{2} (θ) - 5 cos (θ) - 3 = 0

15 sin (x) + 6 cos (x) - 3 = 0

1 + cot^{2} (x) = 8

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

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

What is the general solution for sin(x)+cos^2(x)= 5/4 ?
The general solution for sin(x)+cos^2(x)= 5/4 is x= pi/6+2pin,x=(5pi)/6+2pin