What is the general solution for tan(x)=5sin(x) ?

The general solution for tan(x)=5sin(x) is x=2pin,x=pi+2pin,x=1.36943…+2pin,x=2pi-1.36943…+2pin

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Popular >

tan(x)=5sin(x)

Solution

tan (x) = 5 sin (x)

Solution

x = 2 πn, x = π + 2 πn, x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 78.46304 \dots^{\circ} + 36 0^{\circ} n, x = 281.53695 \dots^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) = 5 sin (x)

Subtract

5 sin (x)

from both sides

tan (x) - 5 sin (x) = 0

Express with sin, cos

tan (x) - 5 sin (x)

Use the basic trigonometric identity:

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

= \frac{sin ( x )}{cos ( x )} - 5 sin (x)

Simplify

\frac{sin ( x )}{cos ( x )} - 5 sin (x) : \frac{sin ( x ) - 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} - 5 sin (x)

Convert element to fraction:

5 sin (x) = \frac{5 sin ( x ) cos ( x )}{cos ( x )}

= \frac{sin ( x )}{cos ( x )} - \frac{5 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 ) - 5 sin ( x ) cos ( x )}{cos ( x )}

= \frac{sin ( x ) - 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x ) - 5 cos ( x ) sin ( x )}{cos ( x )} = 0

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

sin (x) - 5 cos (x) sin (x) = 0

Factor

sin (x) - 5 cos (x) sin (x) : - sin (x) (5 cos (x) - 1)

sin (x) - 5 cos (x) sin (x)

Factor out common term

- sin (x)

= - sin (x) (- 1 + 5 cos (x))

- sin (x) (5 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 5 cos (x) - 1 = 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

5 cos (x) - 1 = 0 : x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

5 cos (x) - 1 = 0

Move

1

to the right side

5 cos (x) - 1 = 0

Add

1

to both sides

5 cos (x) - 1 + 1 = 0 + 1

Simplify

5 cos (x) = 1

5 cos (x) = 1

Divide both sides by

5

5 cos (x) = 1

Divide both sides by

5

\frac{5 cos ( x )}{5} = \frac{1}{5}

Simplify

cos (x) = \frac{1}{5}

cos (x) = \frac{1}{5}

Apply trig inverse properties

cos (x) = \frac{1}{5}

General solutions for

cos (x) = \frac{1}{5}

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

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for tan(x)=5sin(x) ?
The general solution for tan(x)=5sin(x) is x=2pin,x=pi+2pin,x=1.36943…+2pin,x=2pi-1.36943…+2pin