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Popular >

16(l)cos(θ)=5sin(2θ)(2l)

Solution

16 (l) cos (θ) = 5 sin (2 θ) (2 l)

Solution

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 0.92729 \dots + 2 πn, θ = π - 0.92729 \dots + 2 πn

+1

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n, θ = 53.13010 \dots^{\circ} + 36 0^{\circ} n, θ = 126.86989 \dots^{\circ} + 36 0^{\circ} n

Solution steps

16 (l) cos (θ) = 5 sin (2 θ) (2 l)

Subtract

5 sin (2 θ) 2 l

from both sides

16 l cos (θ) - 10 l sin (2 θ) = 0

Rewrite using trig identities

- 10 sin (2 θ) l + 16 cos (θ) l

Use the Double Angle identity:

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

= - 10 l \cdot 2 sin (θ) cos (θ) + 16 l cos (θ)

Simplify

= - 20 l sin (θ) cos (θ) + 16 l cos (θ)

16 cos (θ) l - 20 cos (θ) sin (θ) l = 0

Factor

16 cos (θ) l - 20 cos (θ) sin (θ) l : 4 l cos (θ) (4 - 5 sin (θ))

16 cos (θ) l - 20 cos (θ) sin (θ) l

Rewrite as

= 4 \cdot 4 cos (θ) l - 5 \cdot 4 cos (θ) l sin (θ)

Factor out common term

4 cos (θ) l

= 4 cos (θ) l (4 - 5 sin (θ))

4 l cos (θ) (4 - 5 sin (θ)) = 0

Solving each part separately

cos (θ) = 0 or 4 - 5 sin (θ) = 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

4 - 5 sin (θ) = 0 : θ = arcsin (\frac{4}{5}) + 2 πn, θ = π - arcsin (\frac{4}{5}) + 2 πn

4 - 5 sin (θ) = 0

Move

4

to the right side

4 - 5 sin (θ) = 0

Subtract

4

from both sides

4 - 5 sin (θ) - 4 = 0 - 4

Simplify

- 5 sin (θ) = - 4

- 5 sin (θ) = - 4

Divide both sides by

- 5

- 5 sin (θ) = - 4

Divide both sides by

- 5

\frac{- 5 sin ( θ )}{- 5} = \frac{- 4}{- 5}

Simplify

sin (θ) = \frac{4}{5}

sin (θ) = \frac{4}{5}

Apply trig inverse properties

sin (θ) = \frac{4}{5}

General solutions for

sin (θ) = \frac{4}{5}

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

θ = arcsin (\frac{4}{5}) + 2 πn, θ = π - arcsin (\frac{4}{5}) + 2 πn

θ = arcsin (\frac{4}{5}) + 2 πn, θ = π - arcsin (\frac{4}{5}) + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = arcsin (\frac{4}{5}) + 2 πn, θ = π - arcsin (\frac{4}{5}) + 2 πn

Show solutions in decimal form

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 0.92729 \dots + 2 πn, θ = π - 0.92729 \dots + 2 πn

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