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Popular >

6cos(θ)sin(θ)+cos(θ)=0

Solution

6 cos (θ) sin (θ) + cos (θ) = 0

Solution

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn

+1

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = - 9.59406 \dots^{\circ} + 36 0^{\circ} n, θ = 189.59406 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 cos (θ) sin (θ) + cos (θ) = 0

Factor

6 cos (θ) sin (θ) + cos (θ) : cos (θ) (6 sin (θ) + 1)

6 cos (θ) sin (θ) + cos (θ)

Factor out common term

cos (θ)

= cos (θ) (6 sin (θ) + 1)

cos (θ) (6 sin (θ) + 1) = 0

Solving each part separately

cos (θ) = 0 or 6 sin (θ) + 1 = 0

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

cos (θ) = 0

General solutions for

cos (θ) = 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}

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

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

6 sin (θ) + 1 = 0 : θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn

6 sin (θ) + 1 = 0

Move

1

to the right side

6 sin (θ) + 1 = 0

Subtract

1

from both sides

6 sin (θ) + 1 - 1 = 0 - 1

Simplify

6 sin (θ) = - 1

6 sin (θ) = - 1

Divide both sides by

6

6 sin (θ) = - 1

Divide both sides by

6

\frac{6 sin ( θ )}{6} = \frac{- 1}{6}

Simplify

sin (θ) = - \frac{1}{6}

sin (θ) = - \frac{1}{6}

Apply trig inverse properties

sin (θ) = - \frac{1}{6}

General solutions for

sin (θ) = - \frac{1}{6}

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

θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn

θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn

Show solutions in decimal form

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn

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