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

The general solution for sin(2x)=5cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=1.19028…+pin

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

Solution

sin (2 x) = 5 cos^{2} (x)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 1.19028 \dots + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 68.19859 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 x) = 5 cos^{2} (x)

Subtract

5 cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

cos (x)

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

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

Solving each part separately

cos (x) = 0 or - 5 cos (x) + 2 sin (x) = 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 cos (x) + 2 sin (x) = 0 : x = arctan (\frac{5}{2}) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

\frac{- 5 cos ( x ) + 2 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- 5 + \frac{2 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

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

- 5 + 2 tan (x) = 0

- 5 + 2 tan (x) = 0

Move

5

to the right side

- 5 + 2 tan (x) = 0

Add

5

to both sides

- 5 + 2 tan (x) + 5 = 0 + 5

Simplify

2 tan (x) = 5

2 tan (x) = 5

Divide both sides by

2

2 tan (x) = 5

Divide both sides by

2

\frac{2 tan ( x )}{2} = \frac{5}{2}

Simplify

tan (x) = \frac{5}{2}

tan (x) = \frac{5}{2}

Apply trig inverse properties

tan (x) = \frac{5}{2}

General solutions for

tan (x) = \frac{5}{2}

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

x = arctan (\frac{5}{2}) + πn

x = arctan (\frac{5}{2}) + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arctan (\frac{5}{2}) + πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 1.19028 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(2x)=5cos^2(x) ?
The general solution for sin(2x)=5cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=1.19028…+pin