English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sqrt(3)cot(x)sin(x)+2cos^2(x)=0

Solution

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

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

2 cos^{2} (x) + cot (x) sin (x) 3

Use the basic trigonometric identity:

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

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

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

\frac{cos ( x )}{sin ( x )} sin (x) 3

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= cos (x) 3

= 2 cos^{2} (x) + 3 cos (x)

2 cos^{2} (x) + cos (x) 3 = 0

Solve by substitution

2 cos^{2} (x) + cos (x) 3 = 0

Let:

cos (x) = u

2 u^{2} + u 3 = 0

2 u^{2} + u 3 = 0 : u = 0, u = - \frac{3}{2}

2 u^{2} + u 3 = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} + 3 u = 0

Solve with the quadratic formula

2 u^{2} + 3 u = 0

Quadratic Equation Formula:

For

a = 2, b = 3, c = 0

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

(3)^{2} - 4 \cdot 2 \cdot 0 = 3

(3)^{2} - 4 \cdot 2 \cdot 0

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

4 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 3 - 0

Subtract the numbers:

3 - 0 = 3

= 3

u_{1, 2} = \frac{- 3 \pm 3}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 3 + 3}{2 \cdot 2}, u_{2} = \frac{- 3 - 3}{2 \cdot 2}

u = \frac{- 3 + 3}{2 \cdot 2} : 0

\frac{- 3 + 3}{2 \cdot 2}

Add similar elements:

- 3 + 3 = 0

= \frac{0}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{0}{4}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

u = \frac{- 3 - 3}{2 \cdot 2} : - \frac{3}{2}

\frac{- 3 - 3}{2 \cdot 2}

Add similar elements:

- 3 - 3 = - 23

= \frac{- 2 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2 3}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 3}{4}

Cancel the common factor:

2

= - \frac{3}{2}

The solutions to the quadratic equation are:

u = 0, u = - \frac{3}{2}

Substitute back

u = cos (x)

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

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

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

General solutions for

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Popular Examples

tan^2(x)+sec^2(x)=1

tan^{2} (x) + sec^{2} (x) = 1

1+cos(x)=1-cos(x)

1 + cos (x) = 1 - cos (x)

csc(x)=1-cot(x)

csc (x) = 1 - cot (x)

sin(x)=0.7547

sin (x) = 0.7547

cos(2x-pi/6)=((sqrt(3)))/2 ,0<x<2pi

cos (2 x - \frac{π}{6}) = \frac{( 3 )}{2}, 0 < x < 2 π