What is the general solution for 25sin(2x)-50cos(x)=0 ?

The general solution for 25sin(2x)-50cos(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin

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25sin(2x)-50cos(x)=0

25 sin (2 x) - 50 cos (x) = 0

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

25 sin (2 x) - 50 cos (x) = 0

Rewrite using trig identities

25 sin (2 x) - 50 cos (x)

Use the Double Angle identity:

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

= 25 \cdot 2 sin (x) cos (x) - 50 cos (x)

Simplify

= 50 sin (x) cos (x) - 50 cos (x)

- 50 cos (x) + 50 cos (x) sin (x) = 0

Factor

- 50 cos (x) + 50 cos (x) sin (x) : 50 cos (x) (sin (x) - 1)

- 50 cos (x) + 50 cos (x) sin (x)

Factor out common term

50 cos (x)

= 50 cos (x) (- 1 + sin (x))

50 cos (x) (sin (x) - 1) = 0

Solving each part separately

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

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

cos (x) = 0

General solutions for

cos (x) = 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}

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

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

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

sin (x) - 1 = 0

Move

1

to the right side

sin (x) - 1 = 0

Add

1

to both sides

sin (x) - 1 + 1 = 0 + 1

Simplify

sin (x) = 1

sin (x) = 1

General solutions for

sin (x) = 1

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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

5 sin (2 x) = 3 cos (2 x)

4 cos (3 x - 20) = 1

0 = 1.9 cos (0.1 t) - 1.25 sin (0.2 t)

sin (\frac{x}{4}) = \frac{3}{2}

cos (θ) = \frac{- 3 + ( - 6 )}{( 9 ) ( 18 )}

What is the general solution for 25sin(2x)-50cos(x)=0 ?
The general solution for 25sin(2x)-50cos(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin