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

The general solution for tan(3x)+sec(3x)=2 is x=(0.64350…)/3+(2pin)/3

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

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

x = \frac{0.64350 \dots}{3} + \frac{2 πn}{3}

Solution steps

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

Subtract

2

from both sides

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

Express with sin, cos

\frac{sin ( 3 x )}{cos ( 3 x )} + \frac{1}{cos ( 3 x )} - 2 = 0

Simplify

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

\frac{sin ( 3 x ) + 1 - 2 cos ( 3 x )}{cos ( 3 x )} = 0

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

sin (3 x) + 1 - 2 cos (3 x) = 0

Add

2 cos (3 x)

to both sides

sin (3 x) + 1 = 2 cos (3 x)

Square both sides

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

Subtract

(2 cos (3 x))^{2}

from both sides

(sin (3 x) + 1)^{2} - 4 cos^{2} (3 x) = 0

Rewrite using trig identities

- 3 + 2 sin (3 x) + 5 sin^{2} (3 x) = 0

Solve by substitution

sin (3 x) = \frac{3}{5}, sin (3 x) = - 1

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

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

Combine all the solutions

x = \frac{arcsin ( \frac{3}{5} )}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{arcsin ( \frac{3}{5} )}{3} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

Verify solutions by plugging them into the original equation

x = \frac{arcsin ( \frac{3}{5} )}{3} + \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{0.64350 \dots}{3} + \frac{2 πn}{3}

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

sin (θ) = \frac{- 2}{3}

sec^{2} (θ) + csc^{2} (θ) = 4

cos (\frac{θ}{3}) = sin (θ)

sin (x) - cos (x) = \frac{1}{5}

What is the general solution for tan(3x)+sec(3x)=2 ?
The general solution for tan(3x)+sec(3x)=2 is x=(0.64350…)/3+(2pin)/3