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

The general solution for sin(x)sec(x)=sin(x) is x=2pin,x=pi+2pin

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sin(x)sec(x)=sin(x)

sin (x) sec (x) = sin (x)

x = 2 πn, x = π + 2 πn

Degrees

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

Solution steps

sin (x) sec (x) = sin (x)

Subtract

sin (x)

from both sides

sin (x) sec (x) - sin (x) = 0

Factor

sin (x) sec (x) - sin (x) : sin (x) (sec (x) - 1)

sin (x) sec (x) - sin (x)

Factor out common term

sin (x)

= sin (x) (sec (x) - 1)

sin (x) (sec (x) - 1) = 0

Solving each part separately

sin (x) = 0 or sec (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

sec (x) - 1 = 0 : x = 2 πn

sec (x) - 1 = 0

Move

1

to the right side

sec (x) - 1 = 0

Add

1

to both sides

sec (x) - 1 + 1 = 0 + 1

Simplify

sec (x) = 1

sec (x) = 1

General solutions for

sec (x) = 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 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

Combine all the solutions

x = 2 πn, x = π + 2 πn

cos (x) + 2 = 0

\frac{2 tan ( 3 x )}{1 - tan ^{2} ( 3 x )} = 1

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

cos (2 θ) = - \frac{1}{2}, 0 \leq θ < 2 π

tan (x) tan (2 x) = 1

What is the general solution for sin(x)sec(x)=sin(x) ?
The general solution for sin(x)sec(x)=sin(x) is x=2pin,x=pi+2pin