What is the general solution for 5sin(x)cos(x)=7cos(x) ?

The general solution for 5sin(x)cos(x)=7cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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5sin(x)cos(x)=7cos(x)

5 sin (x) cos (x) = 7 cos (x)

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

5 sin (x) cos (x) = 7 cos (x)

Subtract

7 cos (x)

from both sides

5 sin (x) cos (x) - 7 cos (x) = 0

Factor

5 sin (x) cos (x) - 7 cos (x) : cos (x) (5 sin (x) - 7)

5 sin (x) cos (x) - 7 cos (x)

Factor out common term

cos (x)

= cos (x) (5 sin (x) - 7)

cos (x) (5 sin (x) - 7) = 0

Solving each part separately

cos (x) = 0 or 5 sin (x) - 7 = 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

5 sin (x) - 7 = 0 :

No Solution

5 sin (x) - 7 = 0

Move

7

to the right side

5 sin (x) - 7 = 0

Add

7

to both sides

5 sin (x) - 7 + 7 = 0 + 7

Simplify

5 sin (x) = 7

5 sin (x) = 7

Divide both sides by

5

5 sin (x) = 7

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{7}{5}

Simplify

sin (x) = \frac{7}{5}

sin (x) = \frac{7}{5}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

tan (\frac{θ}{2}) - sin (θ) = 0

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

cos (θ) cos (2 θ) + sin (θ) sin (2 θ) = \frac{2}{2}

sec^{2} (x) - 2 tan (x) = 4

cos (4 x) = cos (2 x)

What is the general solution for 5sin(x)cos(x)=7cos(x) ?
The general solution for 5sin(x)cos(x)=7cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin