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Popular >

4sin(z)cos^2(z)=3sin(z)

Solution

4 sin (z) cos^{2} (z) = 3 sin (z)

Solution

z = 2 πn, z = π + 2 πn, z = \frac{5 π}{6} + 2 πn, z = \frac{7 π}{6} + 2 πn, z = \frac{π}{6} + 2 πn, z = \frac{11 π}{6} + 2 πn

Degrees

z = 0^{\circ} + 36 0^{\circ} n, z = 18 0^{\circ} + 36 0^{\circ} n, z = 15 0^{\circ} + 36 0^{\circ} n, z = 21 0^{\circ} + 36 0^{\circ} n, z = 3 0^{\circ} + 36 0^{\circ} n, z = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

4 sin (z) cos^{2} (z) = 3 sin (z)

Subtract

3 sin (z)

from both sides

4 cos^{2} (z) sin (z) - 3 sin (z) = 0

Factor

4 cos^{2} (z) sin (z) - 3 sin (z) : sin (z) (2 cos (z) + 3) (2 cos (z) - 3)

4 cos^{2} (z) sin (z) - 3 sin (z)

Factor out common term

sin (z)

= sin (z) (4 cos^{2} (z) - 3)

Factor

4 cos^{2} (z) - 3 : (2 cos (z) + 3) (2 cos (z) - 3)

4 cos^{2} (z) - 3

Rewrite

4 cos^{2} (z) - 3

(2 cos (z))^{2} - (3)^{2}

4 cos^{2} (z) - 3

Rewrite

4

2^{2}

= 2^{2} cos^{2} (z) - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= 2^{2} cos^{2} (z) - (3)^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} cos^{2} (z) = (2 cos (z))^{2}

= (2 cos (z))^{2} - (3)^{2}

= (2 cos (z))^{2} - (3)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 cos (z))^{2} - (3)^{2} = (2 cos (z) + 3) (2 cos (z) - 3)

= (2 cos (z) + 3) (2 cos (z) - 3)

= sin (z) (2 cos (z) + 3) (2 cos (z) - 3)

sin (z) (2 cos (z) + 3) (2 cos (z) - 3) = 0

Solving each part separately

sin (z) = 0 or 2 cos (z) + 3 = 0 or 2 cos (z) - 3 = 0

sin (z) = 0 : z = 2 πn, z = π + 2 πn

sin (z) = 0

General solutions for

sin (z) = 0

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}

z = 0 + 2 πn, z = π + 2 πn

z = 0 + 2 πn, z = π + 2 πn

Solve

z = 0 + 2 πn : z = 2 πn

z = 0 + 2 πn

0 + 2 πn = 2 πn

z = 2 πn

z = 2 πn, z = π + 2 πn

2 cos (z) + 3 = 0 : z = \frac{5 π}{6} + 2 πn, z = \frac{7 π}{6} + 2 πn

2 cos (z) + 3 = 0

Move

3

to the right side

2 cos (z) + 3 = 0

Subtract

3

from both sides

2 cos (z) + 3 - 3 = 0 - 3

Simplify

2 cos (z) = - 3

2 cos (z) = - 3

Divide both sides by

2

2 cos (z) = - 3

Divide both sides by

2

\frac{2 cos ( z )}{2} = \frac{- 3}{2}

Simplify

cos (z) = - \frac{3}{2}

cos (z) = - \frac{3}{2}

General solutions for

cos (z) = - \frac{3}{2}

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}

z = \frac{5 π}{6} + 2 πn, z = \frac{7 π}{6} + 2 πn

z = \frac{5 π}{6} + 2 πn, z = \frac{7 π}{6} + 2 πn

2 cos (z) - 3 = 0 : z = \frac{π}{6} + 2 πn, z = \frac{11 π}{6} + 2 πn

2 cos (z) - 3 = 0

Move

3

to the right side

2 cos (z) - 3 = 0

Add

3

to both sides

2 cos (z) - 3 + 3 = 0 + 3

Simplify

2 cos (z) = 3

2 cos (z) = 3

Divide both sides by

2

2 cos (z) = 3

Divide both sides by

2

\frac{2 cos ( z )}{2} = \frac{3}{2}

Simplify

cos (z) = \frac{3}{2}

cos (z) = \frac{3}{2}

General solutions for

cos (z) = \frac{3}{2}

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}

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

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

Combine all the solutions

z = 2 πn, z = π + 2 πn, z = \frac{5 π}{6} + 2 πn, z = \frac{7 π}{6} + 2 πn, z = \frac{π}{6} + 2 πn, z = \frac{11 π}{6} + 2 πn

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