What is the general solution for sin(x)sec(x)=2sin(x),0<= x<= 2pi ?

Q: What is the general solution for sin(x)sec(x)=2sin(x),0<= x<= 2pi ?

The general solution for sin(x)sec(x)=2sin(x),0<= x<= 2pi is x=0,x=pi,x=2pi,x= pi/3 ,x=(5pi)/3

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Popular >

sin(x)sec(x)=2sin(x),0<= x<= 2pi

Solution

sin (x) sec (x) = 2 sin (x), 0 \leq x \leq 2 π

Solution

x = 0, x = π, x = 2 π, x = \frac{π}{3}, x = \frac{5 π}{3}

Degrees

x = 0^{\circ}, x = 18 0^{\circ}, x = 36 0^{\circ}, x = 6 0^{\circ}, x = 30 0^{\circ}

Solution steps

sin (x) sec (x) = 2 sin (x), 0 \leq x \leq 2 π

Subtract

2 sin (x)

from both sides

sin (x) sec (x) - 2 sin (x) = 0

Factor

sin (x) sec (x) - 2 sin (x) : sin (x) (sec (x) - 2)

sin (x) sec (x) - 2 sin (x)

Factor out common term

sin (x)

= sin (x) (sec (x) - 2)

sin (x) (sec (x) - 2) = 0

Solving each part separately

sin (x) = 0 or sec (x) - 2 = 0

sin (x) = 0, 0 \leq x \leq 2 π : x = 0, x = π, x = 2 π

sin (x) = 0, 0 \leq x \leq 2 π

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

Solutions for the range

0 \leq x \leq 2 π

x = 0, x = π, x = 2 π

sec (x) - 2 = 0, 0 \leq x \leq 2 π : x = \frac{π}{3}, x = \frac{5 π}{3}

sec (x) - 2 = 0, 0 \leq x \leq 2 π

Move

2

to the right side

sec (x) - 2 = 0

Add

2

to both sides

sec (x) - 2 + 2 = 0 + 2

Simplify

sec (x) = 2

sec (x) = 2

General solutions for

sec (x) = 2

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 = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{3}, x = \frac{5 π}{3}

Combine all the solutions

x = 0, x = π, x = 2 π, x = \frac{π}{3}, x = \frac{5 π}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)sec(x)=2sin(x),0<= x<= 2pi ?
The general solution for sin(x)sec(x)=2sin(x),0<= x<= 2pi is x=0,x=pi,x=2pi,x= pi/3 ,x=(5pi)/3