What is the general solution for cos(2x+pi/3)=sin(2x) ?

The general solution for cos(2x+pi/3)=sin(2x) is x=(0.26179…)/2+(pin)/2

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Popular >

cos(2x+pi/3)=sin(2x)

Solution

cos (2 x + \frac{π}{3}) = sin (2 x)

Solution

x = \frac{0.26179 \dots}{2} + \frac{πn}{2}

Degrees

x = 7. 5^{\circ} + 9 0^{\circ} n

Solution steps

cos (2 x + \frac{π}{3}) = sin (2 x)

Rewrite using trig identities

cos (2 x + \frac{π}{3}) = sin (2 x)

Rewrite using trig identities

cos (2 x + \frac{π}{3})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 x) cos (\frac{π}{3}) - sin (2 x) sin (\frac{π}{3})

Simplify

cos (2 x) cos (\frac{π}{3}) - sin (2 x) sin (\frac{π}{3}) : \frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x)

cos (2 x) cos (\frac{π}{3}) - sin (2 x) sin (\frac{π}{3})

Simplify

cos (\frac{π}{3}) : \frac{1}{2}

cos (\frac{π}{3})

Use the following trivial identity:

cos (\frac{π}{3}) = \frac{1}{2}

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}

= \frac{1}{2}

= \frac{1}{2} cos (2 x) - sin (\frac{π}{3}) sin (2 x)

Simplify

sin (\frac{π}{3}) : \frac{3}{2}

sin (\frac{π}{3})

Use the following trivial identity:

sin (\frac{π}{3}) = \frac{3}{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}

= \frac{3}{2}

= \frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x)

= \frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x)

\frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x) = sin (2 x)

\frac{1}{2} cos (2 x) - \frac{3}{2} sin (2 x) = sin (2 x)

Subtract

sin (2 x)

from both sides

\frac{1}{2} cos (2 x) + \frac{- 3 - 2}{2} sin (2 x) = 0

Simplify

\frac{1}{2} cos (2 x) + \frac{- 3 - 2}{2} sin (2 x) : \frac{cos ( 2 x ) + ( - 3 - 2 ) sin ( 2 x )}{2}

\frac{1}{2} cos (2 x) + \frac{- 3 - 2}{2} sin (2 x)

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

\frac{1}{2} cos (2 x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ( 2 x )}{2}

Multiply:

1 \cdot cos (2 x) = cos (2 x)

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

\frac{- 3 - 2}{2} sin (2 x) = \frac{( - 3 - 2 ) sin ( 2 x )}{2}

\frac{- 3 - 2}{2} sin (2 x)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( - 3 - 2 ) sin ( 2 x )}{2}

= \frac{cos ( 2 x )}{2} + \frac{( - 2 - 3 ) sin ( 2 x )}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cos ( 2 x ) + ( - 2 - 3 ) sin ( 2 x )}{2}

\frac{cos ( 2 x ) + ( - 3 - 2 ) sin ( 2 x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (2 x) + (- 3 - 2) sin (2 x) = 0

Rewrite using trig identities

cos (2 x) + (- 3 - 2) sin (2 x) = 0

Divide both sides by

cos (2 x), cos (2 x) \neq = 0

\frac{cos ( 2 x ) + ( - 3 - 2 ) sin ( 2 x )}{cos ( 2 x )} = \frac{0}{cos ( 2 x )}

Simplify

1 - \frac{3 sin ( 2 x )}{cos ( 2 x )} - \frac{2 sin ( 2 x )}{cos ( 2 x )} = 0

Use the basic trigonometric identity:

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

1 - (2 + 3) tan (2 x) = 0

1 - (2 + 3) tan (2 x) = 0

Move

1

to the right side

1 - (2 + 3) tan (2 x) = 0

Subtract

1

from both sides

1 - (2 + 3) tan (2 x) - 1 = 0 - 1

Simplify

- (2 + 3) tan (2 x) = - 1

- (2 + 3) tan (2 x) = - 1

Simplify

- (2 + 3) : - 2 - 3

- (2 + 3)

Distribute parentheses

= - (2) - (3)

Apply minus-plus rules

+ (- a) = - a

= - 2 - 3

(- 2 - 3) tan (2 x) = - 1

Divide both sides by

- 2 - 3

(- 2 - 3) tan (2 x) = - 1

Divide both sides by

- 2 - 3

\frac{( - 2 - 3 ) tan ( 2 x )}{- 2 - 3} = \frac{- 1}{- 2 - 3}

Simplify

\frac{( - 2 - 3 ) tan ( 2 x )}{- 2 - 3} = \frac{- 1}{- 2 - 3}

Simplify

\frac{( - 2 - 3 ) tan ( 2 x )}{- 2 - 3} : tan (2 x)

\frac{( - 2 - 3 ) tan ( 2 x )}{- 2 - 3}

Cancel the common factor:

- 2 - 3

= tan (2 x)

Simplify

\frac{- 1}{- 2 - 3} : 2 - 3

\frac{- 1}{- 2 - 3}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 2 - 3 = - (2 + 3)

= \frac{1}{2 + 3}

Rationalize

\frac{1}{2 + 3} : 2 - 3

\frac{1}{2 + 3}

Multiply by the conjugate

\frac{2 - 3}{2 - 3}

= \frac{1 \cdot ( 2 - 3 )}{( 2 + 3 ) ( 2 - 3 )}

1 \cdot (2 - 3) = 2 - 3

(2 + 3) (2 - 3) = 1

(2 + 3) (2 - 3)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 2, b = 3

= 2^{2} - (3)^{2}

Simplify

2^{2} - (3)^{2} : 1

2^{2} - (3)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 4 - 3

Subtract the numbers:

4 - 3 = 1

= 1

= 1

= \frac{2 - 3}{1}

Apply rule

\frac{a}{1} = a

= 2 - 3

= 2 - 3

tan (2 x) = 2 - 3

tan (2 x) = 2 - 3

tan (2 x) = 2 - 3

Apply trig inverse properties

tan (2 x) = 2 - 3

General solutions for

tan (2 x) = 2 - 3

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

2 x = arctan (2 - 3) + πn

2 x = arctan (2 - 3) + πn

Solve

2 x = arctan (2 - 3) + πn : x = \frac{arctan ( 2 - 3 )}{2} + \frac{πn}{2}

2 x = arctan (2 - 3) + πn

Divide both sides by

2

2 x = arctan (2 - 3) + πn

Divide both sides by

2

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

Simplify

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

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

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

Show solutions in decimal form

x = \frac{0.26179 \dots}{2} + \frac{πn}{2}

Graph

Popular Examples

cot(2t)=0

cot (2 t) = 0

tan(θ)=cot(θ),0<= θ<2pi

tan (θ) = cot (θ), 0 \leq θ < 2 π

sec(θ)=1.984

sec (θ) = 1.984

2sin(1/2 x)-sqrt(3)=0

2 sin (\frac{1}{2} x) - 3 = 0

sin(2θ)=sqrt(2)cos(θ)

sin (2 θ) = 2 cos (θ)

Frequently Asked Questions (FAQ)

What is the general solution for cos(2x+pi/3)=sin(2x) ?
The general solution for cos(2x+pi/3)=sin(2x) is x=(0.26179…)/2+(pin)/2