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

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

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Popular >

sin(θ/2)-cos(θ/2)=0

Solution

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

Solution

θ = \frac{π}{2} + 2 πn

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (\frac{θ}{2}) - 1 = 0

tan (\frac{θ}{2}) - 1 = 0

Move

1

to the right side

tan (\frac{θ}{2}) - 1 = 0

Add

1

to both sides

tan (\frac{θ}{2}) - 1 + 1 = 0 + 1

Simplify

tan (\frac{θ}{2}) = 1

tan (\frac{θ}{2}) = 1

General solutions for

tan (\frac{θ}{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{θ}{2} = \frac{π}{4} + πn

\frac{θ}{2} = \frac{π}{4} + πn

Solve

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

\frac{θ}{2} = \frac{π}{4} + πn

Multiply both sides by

2

\frac{θ}{2} = \frac{π}{4} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

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

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

Popular Examples

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sin (x) = \frac{- 3}{5}

3cos^2(a)+sin^2(a)=2

3 cos^{2} (a) + sin^{2} (a) = 2

2cos(6x)+sqrt(3)=0

2 cos (6 x) + 3 = 0

2=2sin(3θ)

2 = 2 sin (3 θ)

sin(pi/3-x)= 1/2

sin (\frac{π}{3} - x) = \frac{1}{2}

Frequently Asked Questions (FAQ)

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