What is the general solution for 2sin(2x)=-cos(2x) ?

The general solution for 2sin(2x)=-cos(2x) is x=-(0.46364…)/2+(pin)/2

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

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

x = - \frac{0.46364 \dots}{2} + \frac{πn}{2}

Degrees

x = - 13.28252 \dots^{\circ} + 9 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

cos (2 x), cos (2 x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

2 tan (2 x) = - 1

2 tan (2 x) = - 1

Divide both sides by

2

2 tan (2 x) = - 1

Divide both sides by

2

\frac{2 tan ( 2 x )}{2} = \frac{- 1}{2}

Simplify

tan (2 x) = - \frac{1}{2}

tan (2 x) = - \frac{1}{2}

Apply trig inverse properties

tan (2 x) = - \frac{1}{2}

General solutions for

tan (2 x) = - \frac{1}{2}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

2 x = arctan (- \frac{1}{2}) + πn

2 x = arctan (- \frac{1}{2}) + πn

Solve

2 x = arctan (- \frac{1}{2}) + πn : x = - \frac{arctan ( \frac{1}{2} )}{2} + \frac{πn}{2}

2 x = arctan (- \frac{1}{2}) + πn

Simplify

arctan (- \frac{1}{2}) + πn : - arctan (\frac{1}{2}) + πn

arctan (- \frac{1}{2}) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{1}{2}) = - arctan (\frac{1}{2})

= - arctan (\frac{1}{2}) + πn

2 x = - arctan (\frac{1}{2}) + πn

Divide both sides by

2

2 x = - arctan (\frac{1}{2}) + πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arctan ( \frac{1}{2} )}{2} + \frac{πn}{2}

Simplify

x = - \frac{arctan ( \frac{1}{2} )}{2} + \frac{πn}{2}

x = - \frac{arctan ( \frac{1}{2} )}{2} + \frac{πn}{2}

x = - \frac{arctan ( \frac{1}{2} )}{2} + \frac{πn}{2}

Show solutions in decimal form

x = - \frac{0.46364 \dots}{2} + \frac{πn}{2}

cos (2 x) + cos (\frac{π}{2} - 2 x) = - 2

cos^{2} (x) = - 1.5

2 arctan (\frac{1}{2}) - arctan (x) = \frac{π}{4}

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

sin (θ) = \frac{13}{85}

What is the general solution for 2sin(2x)=-cos(2x) ?
The general solution for 2sin(2x)=-cos(2x) is x=-(0.46364…)/2+(pin)/2