5sin(4x)=4cos(2x)

Solution

5 sin (4 x) = 4 cos (2 x)

x = \frac{π + 4 πn}{4}, x = \frac{3 π + 4 πn}{4}, x = \frac{0.41151 \dots + 2 πn}{2}, x = \frac{π - 0.41151 \dots + 2 πn}{2}

Solution steps

5 sin (4 x) = 4 cos (2 x)

Subtract

4 cos (2 x)

from both sides

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

Let:

u = 2 x

5 sin (2 u) - 4 cos (u) = 0

Rewrite using trig identities

- 4 cos (u) + 10 cos (u) sin (u) = 0

Factor

- 4 cos (u) + 10 cos (u) sin (u) : 2 cos (u) (5 sin (u) - 2)

2 cos (u) (5 sin (u) - 2) = 0

Solving each part separately

cos (u) = 0 or 5 sin (u) - 2 = 0

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

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

Combine all the solutions

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

Substitute back

u = 2 x

2 x = \frac{π}{2} + 2 πn : x = \frac{π + 4 πn}{4}

2 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π + 4 πn}{4}

2 x = arcsin (\frac{2}{5}) + 2 πn : x = \frac{arcsin ( \frac{2}{5} ) + 2 πn}{2}

2 x = π - arcsin (\frac{2}{5}) + 2 πn : x = \frac{π - arcsin ( \frac{2}{5} ) + 2 πn}{2}

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

Show solutions in decimal form

x = \frac{π + 4 πn}{4}, x = \frac{3 π + 4 πn}{4}, x = \frac{0.41151 \dots + 2 πn}{2}, x = \frac{π - 0.41151 \dots + 2 πn}{2}

arccosh (x) = 1

θ, sin (3 θ) = \frac{1}{2}

0 = - 0.5 sin (\frac{x}{5})

\frac{408.4}{sin ( 13 0 ^{\circ} )} = \frac{200}{sin ( x )}

cot (θ) = - \frac{7}{6}