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

The general solution for sin(x)*cos(x)=2cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=1.10714…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

sin (x) \cdot cos (x) = 2 cos^{2} (x)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 1.10714 \dots + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

2 cos^{2} (x)

from both sides

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

Factor

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

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

Apply exponent rule:

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

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

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

Factor out common term

cos (x)

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

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

Solving each part separately

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

sin (x) - 2 cos (x) = 0 : x = arctan (2) + πn

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 2 = 0

tan (x) - 2 = 0

Move

2

to the right side

tan (x) - 2 = 0

Add

2

to both sides

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

Simplify

tan (x) = 2

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

Combine all the solutions

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

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 1.10714 \dots + πn

Popular Examples

cos(θ)tan(θ)+cot(θ)=csc(θ)

cos (θ) tan (θ) + cot (θ) = csc (θ)

tan^2(x)= 1/(cot(x))

tan^{2} (x) = \frac{1}{cot ( x )}

(sin(θ))/(cos(θ))=sqrt(3)

\frac{sin ( θ )}{cos ( θ )} = 3

48sin^2(x)=48-24cos(x)

48 sin^{2} (x) = 48 - 24 cos (x)

1-4cos(2x)=0

1 - 4 cos (2 x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)*cos(x)=2cos^2(x) ?
The general solution for sin(x)*cos(x)=2cos^2(x) is x= pi/2+2pin,x=(3pi)/2+2pin,x=1.10714…+pin