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

The general solution for cos(x/2)-sin(x/2)=0 is x= pi/2+2pin

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

Solution

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

Solution

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

cos (\frac{x}{2}), cos (\frac{x}{2}) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

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

Simplify

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

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

General solutions for

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

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{x}{2} = \frac{π}{4} + πn

\frac{x}{2} = \frac{π}{4} + πn

Solve

\frac{x}{2} = \frac{π}{4} + πn : x = \frac{π}{2} + 2 πn

\frac{x}{2} = \frac{π}{4} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{4} + πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{4} + 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{4} + 2 πn : \frac{π}{2} + 2 πn

2 \cdot \frac{π}{4} + 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Popular Examples

csc(2x)=-sqrt(2)

csc (2 x) = - 2

3sin(2x)=-1.76

3 sin (2 x) = - 1.76

sin(θ)= 5/12

sin (θ) = \frac{5}{12}

sin(2θ)=(-1)/2

sin (2 θ) = \frac{- 1}{2}

tan(θ)=(-68.1)/(10)

tan (θ) = \frac{- 68.1}{10}

Frequently Asked Questions (FAQ)

What is the general solution for cos(x/2)-sin(x/2)=0 ?
The general solution for cos(x/2)-sin(x/2)=0 is x= pi/2+2pin