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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2tan(θ)+sin(θ)cos(θ)=0

Solution

2 tan (θ) + sin (θ) cos (θ) = 0

Solution

θ = 2 πn, θ = π + 2 πn

Degrees

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

Solution steps

2 tan (θ) + sin (θ) cos (θ) = 0

Express with sin, cos

2 tan (θ) + cos (θ) sin (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

2 \cdot \frac{sin ( θ )}{cos ( θ )} + cos (θ) sin (θ)

Multiply

2 \cdot \frac{sin ( θ )}{cos ( θ )} : \frac{2 sin ( θ )}{cos ( θ )}

2 \cdot \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

= \frac{sin ( θ ) \cdot 2}{cos ( θ )}

= \frac{2 sin ( θ )}{cos ( θ )} + cos (θ) sin (θ)

Convert element to fraction:

cos (θ) sin (θ) = \frac{cos ( θ ) sin ( θ ) cos ( θ )}{cos ( θ )}

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

Since the denominators are equal, combine the fractions:

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

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

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

sin (θ) \cdot 2 + cos (θ) sin (θ) cos (θ)

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

cos (θ) sin (θ) cos (θ)

Apply exponent rule:

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

cos (θ) cos (θ) = cos^{1 + 1} (θ)

= sin (θ) cos^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin (θ) cos^{2} (θ)

= 2 sin (θ) + cos^{2} (θ) sin (θ)

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

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

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

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

2 sin (θ) + cos^{2} (θ) sin (θ) = 0

Factor

2 sin (θ) + cos^{2} (θ) sin (θ) : sin (θ) (cos^{2} (θ) + 2)

2 sin (θ) + cos^{2} (θ) sin (θ)

Factor out common term

sin (θ)

= sin (θ) (2 + cos^{2} (θ))

sin (θ) (cos^{2} (θ) + 2) = 0

Solving each part separately

sin (θ) = 0 or cos^{2} (θ) + 2 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

cos^{2} (θ) + 2 = 0 :

No Solution

cos^{2} (θ) + 2 = 0

Solve by substitution

cos^{2} (θ) + 2 = 0

Let:

cos (θ) = 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 (θ)

cos (θ) = 2 i, cos (θ) = - 2 i

cos (θ) = 2 i, cos (θ) = - 2 i

cos (θ) = 2 i :

No Solution

cos (θ) = 2 i

No Solution

cos (θ) = - 2 i :

No Solution

cos (θ) = - 2 i

No Solution

Combine all the solutions

No Solution

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

Popular Examples

sec(x)= 25/7

sec (x) = \frac{25}{7}

sin(x/2)= 1/2 sin(x)

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

tan(θ)=(-3)/3

tan (θ) = \frac{- 3}{3}

csc(2θ)= 1/2 (sec(θ))(cos(θ))

csc (2 θ) = \frac{1}{2} (sec (θ)) (cos (θ))

sec^2(x)+2sec(x)-8=0

sec^{2} (x) + 2 sec (x) - 8 = 0

Frequently Asked Questions (FAQ)

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