English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2sin(2x)=3tan(x)

Solution

2 sin (2 x) = 3 tan (x)

Solution

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (2 x) = 3 tan (x)

Subtract

3 tan (x)

from both sides

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

Express with sin, cos

2 sin (2 x) - 3 tan (x)

Use the basic trigonometric identity:

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

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

Simplify

2 sin (2 x) - 3 \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( 2 x ) cos ( x ) - 3 sin ( x )}{cos ( x )}

2 sin (2 x) - 3 \cdot \frac{sin ( x )}{cos ( x )}

Multiply

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

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

- 3 sin (x) + 2 cos (x) sin (2 x)

Use the Double Angle identity:

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

= - 3 sin (x) + 2 cos (x) \cdot 2 sin (x) cos (x)

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

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

Multiply the numbers:

2 \cdot 2 = 4

= 4 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)

= 4 sin (x) cos^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

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

= - 3 sin (x) + 4 cos^{2} (x) sin (x)

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

Factor

- 3 sin (x) + 4 cos^{2} (x) sin (x) : sin (x) (2 cos (x) + 3) (2 cos (x) - 3)

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

Factor out common term

sin (x)

= sin (x) (- 3 + 4 cos^{2} (x))

Factor

4 cos^{2} (x) - 3 : (2 cos (x) + 3) (2 cos (x) - 3)

4 cos^{2} (x) - 3

Rewrite

4 cos^{2} (x) - 3

(2 cos (x))^{2} - (3)^{2}

4 cos^{2} (x) - 3

Rewrite

4

2^{2}

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

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (2 cos (x) + 3) (2 cos (x) - 3)

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

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

Solving each part separately

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

2 cos (x) + 3 = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

2 cos (x) + 3 = 0

Move

3

to the right side

2 cos (x) + 3 = 0

Subtract

3

from both sides

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

Simplify

2 cos (x) = - 3

2 cos (x) = - 3

Divide both sides by

2

2 cos (x) = - 3

Divide both sides by

2

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

Simplify

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

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

2 cos (x) - 3 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 cos (x) - 3 = 0

Move

3

to the right side

2 cos (x) - 3 = 0

Add

3

to both sides

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

Simplify

2 cos (x) = 3

2 cos (x) = 3

Divide both sides by

2

2 cos (x) = 3

Divide both sides by

2

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

Simplify

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

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{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Popular Examples

tan(φ)= 1/(sqrt(3))

tan (φ) = \frac{1}{3}

sin(θ)*csc(18)=1

sin (θ) \cdot csc (1 8^{\circ}) = 1

tan^2(x)+csc^2(x)=3

tan^{2} (x) + csc^{2} (x) = 3

solvefor x,y=arctan(x+y)solve for

x, y = arctan (x + y)

cos(x)=-sin^2(x)-1,0<= x<= 2pi

cos (x) = - sin^{2} (x) - 1, 0 \leq x \leq 2 π