What is the general solution for 5sin(2x)=5cos(x) ?

The general solution for 5sin(2x)=5cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin

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

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

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

Subtract

5 cos (x)

from both sides

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

Rewrite using trig identities

- 5 cos (x) + 10 cos (x) sin (x) = 0

Factor

- 5 cos (x) + 10 cos (x) sin (x) : 5 cos (x) (2 sin (x) - 1)

5 cos (x) (2 sin (x) - 1) = 0

Solving each part separately

cos (x) = 0 or 2 sin (x) - 1 = 0

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

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 cos^{2} (x) - 3 cos (x) - 5 = 0

cos (2 x) = - cos (x), 0 \leq x \leq 2 π

2 cos^{2} (w) + 3 cos (w) + 1 = 0

cos (3 x) + sin (3 x) = 1

3 csc (2 x) = - 2

What is the general solution for 5sin(2x)=5cos(x) ?
The general solution for 5sin(2x)=5cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin