sin(4y)+sin(6y)+cos(y)=0

Solution

sin (4 y) + sin (6 y) + cos (y) = 0

y = \frac{π}{2} + 2 πn, y = \frac{3 π}{2} + 2 πn, y = \frac{7 π}{30} + \frac{2 πn}{5}, y = \frac{11 π}{30} + \frac{2 πn}{5}

Solution steps

sin (4 y) + sin (6 y) + cos (y) = 0

Rewrite using trig identities

cos (y) + 2 cos (y) sin (5 y) = 0

Factor

cos (y) + 2 cos (y) sin (5 y) : cos (y) (2 sin (5 y) + 1)

cos (y) (2 sin (5 y) + 1) = 0

Solving each part separately

cos (y) = 0 or 2 sin (5 y) + 1 = 0

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

2 sin (5 y) + 1 = 0 : y = \frac{7 π}{30} + \frac{2 πn}{5}, y = \frac{11 π}{30} + \frac{2 πn}{5}

Combine all the solutions

y = \frac{π}{2} + 2 πn, y = \frac{3 π}{2} + 2 πn, y = \frac{7 π}{30} + \frac{2 πn}{5}, y = \frac{11 π}{30} + \frac{2 πn}{5}

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

x, ln (y) = arctan (x) + C

sin (x) + 2 sin (x) = 5

cos (θ) = - \frac{3}{10}

2 cos^{3} (x) = 0