What is the general solution for 4tan(x)+2sin(x)cos(x)=0 ?

The general solution for 4tan(x)+2sin(x)cos(x)=0 is x=2pin,x=pi+2pin

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Popular Trigonometry >

4tan(x)+2sin(x)cos(x)=0

Solution

4 tan (x) + 2 sin (x) cos (x) = 0

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

4 tan (x) + 2 sin (x) cos (x) = 0

Express with sin, cos

4 tan (x) + 2 cos (x) sin (x)

Use the basic trigonometric identity:

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

= 4 \cdot \frac{sin ( x )}{cos ( x )} + 2 cos (x) sin (x)

Simplify

4 \cdot \frac{sin ( x )}{cos ( x )} + 2 cos (x) sin (x) : \frac{4 sin ( x ) + 2 cos ^{2} ( x ) sin ( x )}{cos ( x )}

4 \cdot \frac{sin ( x )}{cos ( x )} + 2 cos (x) sin (x)

Multiply

4 \cdot \frac{sin ( x )}{cos ( x )} : \frac{4 sin ( x )}{cos ( x )}

4 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( x ) \cdot 4}{cos ( x )}

= \frac{4 sin ( x )}{cos ( x )} + 2 cos (x) sin (x)

Convert element to fraction:

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

= \frac{sin ( x ) \cdot 4}{cos ( x )} + \frac{2 cos ( x ) sin ( x ) cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ( x ) \cdot 4 + 2 cos ( x ) sin ( x ) cos ( x )}{cos ( x )}

sin (x) \cdot 4 + 2 cos (x) sin (x) cos (x) = 4 sin (x) + 2 cos^{2} (x) sin (x)

sin (x) \cdot 4 + 2 cos (x) sin (x) cos (x)

2 cos (x) sin (x) cos (x) = 2 cos^{2} (x) sin (x)

2 cos (x) sin (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

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

Add the numbers:

1 + 1 = 2

= 2 sin (x) cos^{2} (x)

= 4 sin (x) + 2 cos^{2} (x) sin (x)

= \frac{4 sin ( x ) + 2 cos ^{2} ( x ) sin ( x )}{cos ( x )}

= \frac{4 sin ( x ) + 2 cos ^{2} ( x ) sin ( x )}{cos ( x )}

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

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

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

Factor

4 sin (x) + 2 cos^{2} (x) sin (x) : 2 sin (x) (cos^{2} (x) + 2)

4 sin (x) + 2 cos^{2} (x) sin (x)

Rewrite

4

2 \cdot 2

= 2 \cdot 2 sin (x) + 2 sin (x) cos^{2} (x)

Factor out common term

2 sin (x)

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

2 sin (x) (cos^{2} (x) + 2) = 0

Solving each part separately

sin (x) = 0 or cos^{2} (x) + 2 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

cos^{2} (x) + 2 = 0 :

No Solution

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

Solve by substitution

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

Let:

cos (x) = u

u^{2} + 2 = 0

u^{2} + 2 = 0 : u = 2 i, u = - 2 i

u^{2} + 2 = 0

Move

2

to the right side

u^{2} + 2 = 0

Subtract

2

from both sides

u^{2} + 2 - 2 = 0 - 2

Simplify

u^{2} = - 2

u^{2} = - 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 2, u = - - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

Simplify

- - 2 : - 2 i

- - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

Substitute back

u = cos (x)

cos (x) = 2 i, cos (x) = - 2 i

cos (x) = 2 i, cos (x) = - 2 i

cos (x) = 2 i :

No Solution

cos (x) = 2 i

No Solution

cos (x) = - 2 i :

No Solution

cos (x) = - 2 i

No Solution

Combine all the solutions

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Graph

Popular Examples

arccot(x-2)=arccot(x-1)+arccot(x)

Frequently Asked Questions (FAQ)

What is the general solution for 4tan(x)+2sin(x)cos(x)=0 ?
The general solution for 4tan(x)+2sin(x)cos(x)=0 is x=2pin,x=pi+2pin