What is the general solution for sin(2x)+3cos(2x)=0 ?

The general solution for sin(2x)+3cos(2x)=0 is x=-(1.24904…)/2+(pin)/2

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

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

x = - \frac{1.24904 \dots}{2} + \frac{πn}{2}

Degrees

x = - 35.78252 \dots^{\circ} + 9 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (2 x) + 3 = 0

tan (2 x) + 3 = 0

Move

3

to the right side

tan (2 x) + 3 = 0

Subtract

3

from both sides

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

Simplify

tan (2 x) = - 3

tan (2 x) = - 3

Apply trig inverse properties

tan (2 x) = - 3

General solutions for

tan (2 x) = - 3

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

2 x = arctan (- 3) + πn

2 x = arctan (- 3) + πn

Solve

2 x = arctan (- 3) + πn : x = - \frac{arctan ( 3 )}{2} + \frac{πn}{2}

2 x = arctan (- 3) + πn

Simplify

arctan (- 3) + πn : - arctan (3) + πn

arctan (- 3) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 3) = - arctan (3)

= - arctan (3) + πn

2 x = - arctan (3) + πn

Divide both sides by

2

2 x = - arctan (3) + πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arctan ( 3 )}{2} + \frac{πn}{2}

Simplify

x = - \frac{arctan ( 3 )}{2} + \frac{πn}{2}

x = - \frac{arctan ( 3 )}{2} + \frac{πn}{2}

x = - \frac{arctan ( 3 )}{2} + \frac{πn}{2}

Show solutions in decimal form

x = - \frac{1.24904 \dots}{2} + \frac{πn}{2}

7 cos (x) tan (x) - 5 tan (x) = 0

sec^{2} (x) - 6 sec (x) = 0

9 sec^{2} (x) tan (x) = 12 tan (x)

cos (x) = - \frac{4}{7}

(8 cos (θ) - 4) (5 sin (θ) - 5) = 0

What is the general solution for sin(2x)+3cos(2x)=0 ?
The general solution for sin(2x)+3cos(2x)=0 is x=-(1.24904…)/2+(pin)/2