What is the general solution for 2cos(x)+sec(x)=3 ?

The general solution for 2cos(x)+sec(x)=3 is x= pi/3+2pin,x=(5pi)/3+2pin,x=2pin

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2cos(x)+sec(x)=3

2 cos (x) + sec (x) = 3

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 2 πn

Solution steps

2 cos (x) + sec (x) = 3

Subtract

3

from both sides

2 cos (x) + sec (x) - 3 = 0

Rewrite using trig identities

- 3 + \frac{2}{sec ( x )} + sec (x) = 0

Solve by substitution

sec (x) = 2, sec (x) = 1

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = 1 : x = 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = 2 πn

cos^{2} (2 x) = 0

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

2 cos (x) - sin (2 x) = 0

4 sin^{2} (x) - 1 = 0, 0 \leq x \leq \frac{π}{2}

\frac{2}{3} = cos (x)

What is the general solution for 2cos(x)+sec(x)=3 ?
The general solution for 2cos(x)+sec(x)=3 is x= pi/3+2pin,x=(5pi)/3+2pin,x=2pin