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Popular Trigonometry >

sin(2x)+1.5cos(x)=0

Solution

sin (2 x) + 1.5 cos (x) = 0

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn

+1

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 228.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (2 x) + 1.5 cos (x) = 0

Rewrite using trig identities

sin (2 x) + 1.5 cos (x)

Use the Double Angle identity:

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

= 2 sin (x) cos (x) + 1.5 cos (x)

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

Factor

1.5 cos (x) + 2 cos (x) sin (x) : cos (x) (2 sin (x) + 1.5)

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

Factor out common term

cos (x)

= cos (x) (1.5 + 2 sin (x))

cos (x) (2 sin (x) + 1.5) = 0

Solving each part separately

cos (x) = 0 or 2 sin (x) + 1.5 = 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

2 sin (x) + 1.5 = 0 : x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

2 sin (x) + 1.5 = 0

Move

1.5

to the right side

2 sin (x) + 1.5 = 0

Subtract

1.5

from both sides

2 sin (x) + 1.5 - 1.5 = 0 - 1.5

Simplify

2 sin (x) = - 1.5

2 sin (x) = - 1.5

Divide both sides by

2

2 sin (x) = - 1.5

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{- 1.5}{2}

Simplify

\frac{2 sin ( x )}{2} = \frac{- 1.5}{2}

Simplify

\frac{2 sin ( x )}{2} : sin (x)

\frac{2 sin ( x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= sin (x)

Simplify

\frac{- 1.5}{2} : - 0.75

\frac{- 1.5}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1.5}{2}

Divide the numbers:

\frac{1.5}{2} = 0.75

= - 0.75

sin (x) = - 0.75

sin (x) = - 0.75

sin (x) = - 0.75

Apply trig inverse properties

sin (x) = - 0.75

General solutions for

sin (x) = - 0.75

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

x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (- 0.75) + 2 πn, x = π + arcsin (0.75) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn

Graph

Popular Examples

2sin(2x)=sin^2(x)

sin^2(x)=-(1/2)

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