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

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

The general solution for sin(2x)=-cos(x),0<= x<= 2pi is x= pi/2 ,x=(3pi)/2 ,x=(7pi)/6 ,x=(11pi)/6

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sin(2x)=-cos(x),0<= x<= 2pi

Solution

sin (2 x) = - cos (x), 0 \leq x \leq 2 π

Solution

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{7 π}{6}, x = \frac{11 π}{6}

Degrees

x = 9 0^{\circ}, x = 27 0^{\circ}, x = 21 0^{\circ}, x = 33 0^{\circ}

Solution steps

sin (2 x) = - cos (x), 0 \leq x \leq 2 π

Subtract

- cos (x)

from both sides

sin (2 x) + cos (x) = 0

Rewrite using trig identities

cos (x) + sin (2 x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

cos (x) + 2 cos (x) sin (x) = 0

Factor

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

cos (x) + 2 cos (x) sin (x)

Factor out common term

cos (x)

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

cos (x) (2 sin (x) + 1) = 0

Solving each part separately

cos (x) = 0 or 2 sin (x) + 1 = 0

cos (x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}

cos (x) = 0, 0 \leq x \leq 2 π

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

2 sin (x) + 1 = 0, 0 \leq x \leq 2 π : x = \frac{7 π}{6}, x = \frac{11 π}{6}

2 sin (x) + 1 = 0, 0 \leq x \leq 2 π

Move

1

to the right side

2 sin (x) + 1 = 0

Subtract

1

from both sides

2 sin (x) + 1 - 1 = 0 - 1

Simplify

2 sin (x) = - 1

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{- 1}{2}

Simplify

sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{2}

General solutions for

sin (x) = - \frac{1}{2}

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 = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solutions for the range

0 \leq x \leq 2 π

x = \frac{7 π}{6}, x = \frac{11 π}{6}

Combine all the solutions

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{7 π}{6}, x = \frac{11 π}{6}

Popular Examples

sin(θ)=(5pi)/6

sin (θ) = \frac{5 π}{6}

sin(x)=(4.2)/(9.9)

sin (x) = \frac{4.2}{9.9}

sin(x)-0.5cos(x)=0.1

sin (x) - 0.5 cos (x) = 0.1

cot(θ)=0.8701

cot (θ) = 0.8701

cos(x)=-sqrt(1+576/625)

cos (x) = - 1 + \frac{576}{625}

Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=-cos(x),0<= x<= 2pi ?
The general solution for sin(2x)=-cos(x),0<= x<= 2pi is x= pi/2 ,x=(3pi)/2 ,x=(7pi)/6 ,x=(11pi)/6