What is the general solution for 5sin^2(x)-6sin(x)cos(x)=0 ?

The general solution for 5sin^2(x)-6sin(x)cos(x)=0 is x=2pin,x=pi+2pin,x=0.87605…+pin

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Popular >

5sin^2(x)-6sin(x)cos(x)=0

Solution

5 sin^{2} (x) - 6 sin (x) cos (x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = 0.87605 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 50.19442 \dots^{\circ} + 18 0^{\circ} n

Solution steps

5 sin^{2} (x) - 6 sin (x) cos (x) = 0

Factor

5 sin^{2} (x) - 6 sin (x) cos (x) : sin (x) (5 sin (x) - 6 cos (x))

5 sin^{2} (x) - 6 sin (x) cos (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

= 5 sin (x) sin (x) - 6 sin (x) cos (x)

Factor out common term

sin (x)

= sin (x) (5 sin (x) - 6 cos (x))

sin (x) (5 sin (x) - 6 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 5 sin (x) - 6 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

5 sin (x) - 6 cos (x) = 0 : x = arctan (\frac{6}{5}) + πn

5 sin (x) - 6 cos (x) = 0

Rewrite using trig identities

5 sin (x) - 6 cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{5 sin ( x ) - 6 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{5 sin ( x )}{cos ( x )} - 6 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

5 tan (x) - 6 = 0

5 tan (x) - 6 = 0

Move

6

to the right side

5 tan (x) - 6 = 0

Add

6

to both sides

5 tan (x) - 6 + 6 = 0 + 6

Simplify

5 tan (x) = 6

5 tan (x) = 6

Divide both sides by

5

5 tan (x) = 6

Divide both sides by

5

\frac{5 tan ( x )}{5} = \frac{6}{5}

Simplify

tan (x) = \frac{6}{5}

tan (x) = \frac{6}{5}

Apply trig inverse properties

tan (x) = \frac{6}{5}

General solutions for

tan (x) = \frac{6}{5}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{6}{5}) + πn

x = arctan (\frac{6}{5}) + πn

Combine all the solutions

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

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.87605 \dots + πn

Popular Examples

10sin(20t-pi/6)=8

10 sin (20 t - \frac{π}{6}) = 8

solvefor x,-cos(x)=0 solve for

x, - cos (x) = 0

sin^2(x)tan^2(x)=tan^2(x)+sin^2(x)

sin^{2} (x) tan^{2} (x) = tan^{2} (x) + sin^{2} (x)

solvefor θ,cot(θ)=-pi/6 solve for

θ, cot (θ) = - \frac{π}{6}

tan(x)+cos(x)=0

tan (x) + cos (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 5sin^2(x)-6sin(x)cos(x)=0 ?
The general solution for 5sin^2(x)-6sin(x)cos(x)=0 is x=2pin,x=pi+2pin,x=0.87605…+pin