What is the general solution for 2tan(x)sin(x)-7sin(x)=0 ?

The general solution for 2tan(x)sin(x)-7sin(x)=0 is x=2pin,x=pi+2pin,x=1.29249…+pin

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2tan(x)sin(x)-7sin(x)=0

2 tan (x) sin (x) - 7 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 1.29249 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 74.05460 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 tan (x) sin (x) - 7 sin (x) = 0

Factor

2 tan (x) sin (x) - 7 sin (x) : sin (x) (2 tan (x) - 7)

2 tan (x) sin (x) - 7 sin (x)

Factor out common term

sin (x)

= sin (x) (2 tan (x) - 7)

sin (x) (2 tan (x) - 7) = 0

Solving each part separately

sin (x) = 0 or 2 tan (x) - 7 = 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 tan (x) - 7 = 0 : x = arctan (\frac{7}{2}) + πn

2 tan (x) - 7 = 0

Move

7

to the right side

2 tan (x) - 7 = 0

Add

7

to both sides

2 tan (x) - 7 + 7 = 0 + 7

Simplify

2 tan (x) = 7

2 tan (x) = 7

Divide both sides by

2

2 tan (x) = 7

Divide both sides by

2

\frac{2 tan ( x )}{2} = \frac{7}{2}

Simplify

tan (x) = \frac{7}{2}

tan (x) = \frac{7}{2}

Apply trig inverse properties

tan (x) = \frac{7}{2}

General solutions for

tan (x) = \frac{7}{2}

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

x = arctan (\frac{7}{2}) + πn

x = arctan (\frac{7}{2}) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (\frac{7}{2}) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.29249 \dots + πn

sin^{2} (x) + sin (x) = 2

- 3 sin^{2} (θ) + 3 = - 10 cos (θ) - 3

cos (2 x) + 2 sin (x) = 1

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

tanh (z) = 0

What is the general solution for 2tan(x)sin(x)-7sin(x)=0 ?
The general solution for 2tan(x)sin(x)-7sin(x)=0 is x=2pin,x=pi+2pin,x=1.29249…+pin