What is the general solution for 4sin(x)-sec(x)=0 ?

The general solution for 4sin(x)-sec(x)=0 is x= pi/(12)+pin,x=(5pi)/(12)+pin

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4sin(x)-sec(x)=0

4 sin (x) - sec (x) = 0

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Solution steps

4 sin (x) - sec (x) = 0

Express with sin, cos

\frac{- 1 + 4 cos ( x ) sin ( x )}{cos ( x )} = 0

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

- 1 + 4 cos (x) sin (x) = 0

Rewrite using trig identities

- 1 + 4 \cdot \frac{sin ( 2 x )}{2} = 0

4 \cdot \frac{sin ( 2 x )}{2} = 2 sin (2 x)

- 1 + 2 sin (2 x) = 0

Move

1

to the right side

2 sin (2 x) = 1

Divide both sides by

2

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

General solutions for

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

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

Solve

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

Solve

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

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

cos (x) = - \frac{1}{2}, 0 < x < 2 π

cot (θ) = cos (θ)

3 sin^{2} (θ) - cos (θ) + 1 = 0

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

What is the general solution for 4sin(x)-sec(x)=0 ?
The general solution for 4sin(x)-sec(x)=0 is x= pi/(12)+pin,x=(5pi)/(12)+pin