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Popular Trigonometry >

-2cos(x)-4sin(2x)=0

Solution

- 2 cos (x) - 4 sin (2 x) = 0

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

+1

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 2 cos (x) - 4 sin (2 x) = 0

Rewrite using trig identities

- 2 cos (x) - 4 sin (2 x)

Use the Double Angle identity:

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

= - 2 cos (x) - 4 \cdot 2 sin (x) cos (x)

Simplify

= - 2 cos (x) - 8 sin (x) cos (x)

- 2 cos (x) - 8 cos (x) sin (x) = 0

Factor

- 2 cos (x) - 8 cos (x) sin (x) : - 2 cos (x) (4 sin (x) + 1)

- 2 cos (x) - 8 cos (x) sin (x)

Rewrite

- 8

as

4 \cdot 2

= - 2 cos (x) + 4 \cdot 2 sin (x) cos (x)

Factor out common term

- 2 cos (x)

= - 2 cos (x) (1 + 4 sin (x))

- 2 cos (x) (4 sin (x) + 1) = 0

Solving each part separately

cos (x) = 0 or 4 sin (x) + 1 = 0

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

cos (x) = 0

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

4 sin (x) + 1 = 0 : x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

4 sin (x) + 1 = 0

Move

1

to the right side

4 sin (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

4 sin (x) = - 1

4 sin (x) = - 1

Divide both sides by

4

4 sin (x) = - 1

Divide both sides by

4

\frac{4 sin ( x )}{4} = \frac{- 1}{4}

Simplify

sin (x) = - \frac{1}{4}

sin (x) = - \frac{1}{4}

Apply trig inverse properties

sin (x) = - \frac{1}{4}

General solutions for

sin (x) = - \frac{1}{4}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Graph

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