What is the general solution for 4sin(x)=sin(x)tan(x) ?

The general solution for 4sin(x)=sin(x)tan(x) is x=2pin,x=pi+2pin,x=1.32581…+pin

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4sin(x)=sin(x)tan(x)

4 sin (x) = sin (x) tan (x)

x = 2 πn, x = π + 2 πn, x = 1.32581 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 75.96375 \dots^{\circ} + 18 0^{\circ} n

Solution steps

4 sin (x) = sin (x) tan (x)

Subtract

sin (x) tan (x)

from both sides

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

Factor

4 sin (x) - sin (x) tan (x) : - sin (x) (tan (x) - 4)

4 sin (x) - sin (x) tan (x)

Factor out common term

- sin (x)

= - sin (x) (- 4 + tan (x))

- sin (x) (tan (x) - 4) = 0

Solving each part separately

sin (x) = 0 or tan (x) - 4 = 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) - 4 = 0 : x = arctan (4) + πn

tan (x) - 4 = 0

Move

4

to the right side

tan (x) - 4 = 0

Add

4

to both sides

tan (x) - 4 + 4 = 0 + 4

Simplify

tan (x) = 4

tan (x) = 4

Apply trig inverse properties

tan (x) = 4

General solutions for

tan (x) = 4

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

x = arctan (4) + πn

x = arctan (4) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (4) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.32581 \dots + πn

cos (x) = 0.707

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

cot (x) = \frac{4}{3}

tan (θ) = - \frac{5}{2}

- 1 = cos (t) + cos (2 t)

What is the general solution for 4sin(x)=sin(x)tan(x) ?
The general solution for 4sin(x)=sin(x)tan(x) is x=2pin,x=pi+2pin,x=1.32581…+pin