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

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

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

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

x = \frac{3 π}{2} + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

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{3 π}{4} + πn

\frac{x}{2} = \frac{3 π}{4} + πn

Solve

\frac{x}{2} = \frac{3 π}{4} + πn : x = \frac{3 π}{2} + 2 πn

\frac{x}{2} = \frac{3 π}{4} + πn

Multiply both sides by

2

\frac{x}{2} = \frac{3 π}{4} + πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{3 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{3 π}{4} + 2 πn : \frac{3 π}{2} + 2 πn

2 \cdot \frac{3 π}{4} + 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

2 cos^{2} (2 x) - 1 = 0

5 cos (x) cot (x) - 3 cos (x) - 10 cot (x) + 6 = 0

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

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

y, cos (y) = x

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