What is the general solution for 4sin(θ/2)=4cos(θ/2) ?

The general solution for 4sin(θ/2)=4cos(θ/2) is θ= pi/2+2pin

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4sin(θ/2)=4cos(θ/2)

4 sin (\frac{θ}{2}) = 4 cos (\frac{θ}{2})

θ = \frac{π}{2} + 2 πn

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

4 sin (\frac{θ}{2}) = 4 cos (\frac{θ}{2})

Rewrite using trig identities

4 sin (\frac{θ}{2}) = 4 cos (\frac{θ}{2})

Divide both sides by

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

\frac{4 sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )} = \frac{4 cos ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )}

Simplify

\frac{4 sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )} = 4

Use the basic trigonometric identity:

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

4 tan (\frac{θ}{2}) = 4

4 tan (\frac{θ}{2}) = 4

Divide both sides by

4

4 tan (\frac{θ}{2}) = 4

Divide both sides by

4

\frac{4 tan ( \frac{θ}{2} )}{4} = \frac{4}{4}

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

2 sec (x) + 4 = sec (x) + 6

sin^{2} (x) - 5 sin (x) = 0

7 tan (c) + 7 = 2 tan (c) + 1

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

sin (x - \frac{π}{2}) = - \frac{2}{2}

What is the general solution for 4sin(θ/2)=4cos(θ/2) ?
The general solution for 4sin(θ/2)=4cos(θ/2) is θ= pi/2+2pin