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

The general solution for sin(x)tan(x)=3sin(x) is x=2pin,x=pi+2pin,x=1.24904…+pin

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

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

x = 2 πn, x = π + 2 πn, x = 1.24904 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

3 sin (x)

from both sides

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

Factor

sin (x) tan (x) - 3 sin (x) : sin (x) (tan (x) - 3)

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

Factor out common term

sin (x)

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

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

Solving each part separately

sin (x) = 0 or tan (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

tan (x) - 3 = 0 : x = arctan (3) + πn

tan (x) - 3 = 0

Move

3

to the right side

tan (x) - 3 = 0

Add

3

to both sides

tan (x) - 3 + 3 = 0 + 3

Simplify

tan (x) = 3

tan (x) = 3

Apply trig inverse properties

tan (x) = 3

General solutions for

tan (x) = 3

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (3) + πn

x = arctan (3) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (3) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.24904 \dots + πn

12 tan (θ) + 5 = 5 tan (θ) + 5

1 - sin (x) - 2 sin^{2} (x) = 0

sin^{2} (x) + cos^{2} (x) - 1 + cos (x) - sin (x) = 0

sin (4 x) - cos (2 x) = 0

\frac{6}{π} = 3 cos (2 x)

What is the general solution for sin(x)tan(x)=3sin(x) ?
The general solution for sin(x)tan(x)=3sin(x) is x=2pin,x=pi+2pin,x=1.24904…+pin