What is the general solution for 5cos(x)-10sin(x)cos(x)=0,0<= x<= pi/2 ?

Q: What is the general solution for 5cos(x)-10sin(x)cos(x)=0,0<= x<= pi/2 ?

The general solution for 5cos(x)-10sin(x)cos(x)=0,0<= x<= pi/2 is x= pi/2 ,x= pi/6

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5cos(x)-10sin(x)cos(x)=0,0<= x<= pi/2

Solution

5 cos (x) - 10 sin (x) cos (x) = 0, 0 \leq x \leq \frac{π}{2}

Solution

x = \frac{π}{2}, x = \frac{π}{6}

Degrees

x = 9 0^{\circ}, x = 3 0^{\circ}

Solution steps

5 cos (x) - 10 sin (x) cos (x) = 0, 0 \leq x \leq \frac{π}{2}

Factor

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

5 cos (x) - 10 sin (x) cos (x)

Rewrite

- 10

2 \cdot 5

= 5 cos (x) + 2 \cdot 5 sin (x) cos (x)

Factor out common term

- 5 cos (x)

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

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

Solving each part separately

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

cos (x) = 0, 0 \leq x \leq \frac{π}{2} : x = \frac{π}{2}

cos (x) = 0, 0 \leq x \leq \frac{π}{2}

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

Solutions for the range

0 \leq x \leq \frac{π}{2}

x = \frac{π}{2}

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

2 sin (x) - 1 = 0, 0 \leq x \leq \frac{π}{2}

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to both sides

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

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

Solutions for the range

0 \leq x \leq \frac{π}{2}

x = \frac{π}{6}

Combine all the solutions

x = \frac{π}{2}, x = \frac{π}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for 5cos(x)-10sin(x)cos(x)=0,0<= x<= pi/2 ?
The general solution for 5cos(x)-10sin(x)cos(x)=0,0<= x<= pi/2 is x= pi/2 ,x= pi/6