What is the general solution for -2sin(x/2)-1=-2cos(x/2)-1 ?

The general solution for -2sin(x/2)-1=-2cos(x/2)-1 is x= pi/2+2pin

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

Solution

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

Solution

x = \frac{π}{2} + 2 πn

Degrees

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

Solution steps

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

Subtract

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

from both sides

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

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

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

Move

2

to the right side

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

Subtract

2

from both sides

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

Simplify

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

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

Divide both sides by

- 2

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

Divide both sides by

- 2

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

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

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

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

tan(θ)=30.48(80)

tan (θ) = 30.48 (80)

cos(x)=a,(3pi)/2 <= x<= 2pi

cos (x) = a, \frac{3 π}{2} \leq x \leq 2 π

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

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

cos(x)=0.77

cos (x) = 0.77

Frequently Asked Questions (FAQ)

What is the general solution for -2sin(x/2)-1=-2cos(x/2)-1 ?
The general solution for -2sin(x/2)-1=-2cos(x/2)-1 is x= pi/2+2pin