What is the general solution for tan(1/2 x)=3cos(1/2 x) ?

The general solution for tan(1/2 x)=3cos(1/2 x) is x=2*1.01055…+4pin,x=2pi-2*1.01055…+4pin

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tan(1/2 x)=3cos(1/2 x)

tan (\frac{1}{2} x) = 3 cos (\frac{1}{2} x)

x = 2 \cdot 1.01055 \dots + 4 πn, x = 2 π - 2 \cdot 1.01055 \dots + 4 πn

Solution steps

tan (\frac{1}{2} x) = 3 cos (\frac{1}{2} x)

Subtract

3 cos (\frac{1}{2} x)

from both sides

tan (\frac{x}{2}) - 3 cos (\frac{x}{2}) = 0

Express with sin, cos

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

Simplify

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

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

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

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

Add

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

to both sides

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

Square both sides

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

Subtract

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

from both sides

sin^{2} (\frac{x}{2}) - 9 cos^{4} (\frac{x}{2}) = 0

Factor

sin^{2} (\frac{x}{2}) - 9 cos^{4} (\frac{x}{2}) : (sin (\frac{x}{2}) + 3 cos^{2} (\frac{x}{2})) (sin (\frac{x}{2}) - 3 cos^{2} (\frac{x}{2}))

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

Solving each part separately

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

sin (\frac{x}{2}) + 3 cos^{2} (\frac{x}{2}) = 0 : x = - 2 arcsin (\frac{37 - 1}{6}) + 4 πn, x = 2 π + 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn

sin (\frac{x}{2}) - 3 cos^{2} (\frac{x}{2}) = 0 : x = 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn, x = 2 π - 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn

Combine all the solutions

x = - 2 arcsin (\frac{37 - 1}{6}) + 4 πn, x = 2 π + 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn, x = 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn, x = 2 π - 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn

Verify solutions by plugging them into the original equation

x = 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn, x = 2 π - 2 arcsin (\frac{- 1 + 37}{6}) + 4 πn

Show solutions in decimal form

x = 2 \cdot 1.01055 \dots + 4 πn, x = 2 π - 2 \cdot 1.01055 \dots + 4 πn

cos (2 θ) = - \frac{1}{2}, 0 \leq x \leq 2 π

cos (x) = - \frac{5}{13}

sin^{2} (x) = 6 (cos (x) + 1)

2 cos^{2} (x) - cos (x) = 1, 0 \leq x \leq 2 π

7 sin^{2} (θ) - 36 sin (θ) + 5 = 0

What is the general solution for tan(1/2 x)=3cos(1/2 x) ?
The general solution for tan(1/2 x)=3cos(1/2 x) is x=2*1.01055…+4pin,x=2pi-2*1.01055…+4pin