What is the general solution for 2tan(x)-3sin(x)=0 ?

The general solution for 2tan(x)-3sin(x)=0 is x=2pin,x=pi+2pin,x=0.84106…+2pin,x=2pi-0.84106…+2pin

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Express with sin, cos

2 tan (x) - 3 sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

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

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

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

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

Factor

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

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

Factor out common term

- sin (x)

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

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

Solving each part separately

sin (x) = 0 or 3 cos (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

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

3 cos (x) - 2 = 0

Move

2

to the right side

3 cos (x) - 2 = 0

Add

2

to both sides

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

Simplify

3 cos (x) = 2

3 cos (x) = 2

Divide both sides by

3

3 cos (x) = 2

Divide both sides by

3

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2tan(x)-3sin(x)=0 ?
The general solution for 2tan(x)-3sin(x)=0 is x=2pin,x=pi+2pin,x=0.84106…+2pin,x=2pi-0.84106…+2pin