What is the general solution for cos(2θ)-sin(θ)cos(θ)=0 ?

The general solution for cos(2θ)-sin(θ)cos(θ)=0 is θ=(1.10714…)/2+(pin)/2

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Popular >

cos(2θ)-sin(θ)cos(θ)=0

Solution

cos (2 θ) - sin (θ) cos (θ) = 0

Solution

θ = \frac{1.10714 \dots}{2} + \frac{πn}{2}

Degrees

θ = 31.71747 \dots^{\circ} + 9 0^{\circ} n

Solution steps

cos (2 θ) - sin (θ) cos (θ) = 0

Rewrite using trig identities

cos (2 θ) - cos (θ) sin (θ)

Use the Double Angle identity:

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

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

= cos (2 θ) - \frac{sin ( 2 θ )}{2}

Divide both sides by

cos (2 θ), cos (2 θ) \neq = 0

\frac{cos ( 2 θ ) - \frac{s i n ( 2 θ )}{2}}{cos ( 2 θ )} = \frac{0}{cos ( 2 θ )}

Simplify

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

Use the basic trigonometric identity:

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

1 - \frac{tan ( 2 θ )}{2} = 0

1 - \frac{tan ( 2 θ )}{2} = 0

Move

1

to the right side

1 - \frac{tan ( 2 θ )}{2} = 0

Subtract

1

from both sides

1 - \frac{tan ( 2 θ )}{2} - 1 = 0 - 1

Simplify

- \frac{tan ( 2 θ )}{2} = - 1

- \frac{tan ( 2 θ )}{2} = - 1

Multiply both sides by

2

- \frac{tan ( 2 θ )}{2} = - 1

Multiply both sides by

2

2 (- \frac{tan ( 2 θ )}{2}) = 2 (- 1)

Simplify

- tan (2 θ) = - 2

- tan (2 θ) = - 2

Divide both sides by

- 1

- tan (2 θ) = - 2

Divide both sides by

- 1

\frac{- tan ( 2 θ )}{- 1} = \frac{- 2}{- 1}

Simplify

tan (2 θ) = 2

tan (2 θ) = 2

Apply trig inverse properties

tan (2 θ) = 2

General solutions for

tan (2 θ) = 2

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

2 θ = arctan (2) + πn

2 θ = arctan (2) + πn

Solve

2 θ = arctan (2) + πn : θ = \frac{arctan ( 2 )}{2} + \frac{πn}{2}

2 θ = arctan (2) + πn

Divide both sides by

2

2 θ = arctan (2) + πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{arctan ( 2 )}{2} + \frac{πn}{2}

Simplify

θ = \frac{arctan ( 2 )}{2} + \frac{πn}{2}

θ = \frac{arctan ( 2 )}{2} + \frac{πn}{2}

θ = \frac{arctan ( 2 )}{2} + \frac{πn}{2}

Show solutions in decimal form

θ = \frac{1.10714 \dots}{2} + \frac{πn}{2}

Graph

Popular Examples

(cos(x))/2-1/2 =0

\frac{cos ( x )}{2} - \frac{1}{2} = 0

tan(x)= 4/7

tan (x) = \frac{4}{7}

sin(x)=-8/9

sin (x) = - \frac{8}{9}

cos^2(θ)+2cos(θ)=3

cos^{2} (θ) + 2 cos (θ) = 3

cos^2(x)= 1/9

cos^{2} (x) = \frac{1}{9}

Frequently Asked Questions (FAQ)

What is the general solution for cos(2θ)-sin(θ)cos(θ)=0 ?
The general solution for cos(2θ)-sin(θ)cos(θ)=0 is θ=(1.10714…)/2+(pin)/2