What is the general solution for 6sin(θ/2)=-6cos(θ/2) ?

The general solution for 6sin(θ/2)=-6cos(θ/2) is θ=(3pi)/2+2pin

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6sin(θ/2)=-6cos(θ/2)

6 sin (\frac{θ}{2}) = - 6 cos (\frac{θ}{2})

θ = \frac{3 π}{2} + 2 πn

Degrees

θ = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

6 sin (\frac{θ}{2}) = - 6 cos (\frac{θ}{2})

Rewrite using trig identities

6 sin (\frac{θ}{2}) = - 6 cos (\frac{θ}{2})

Divide both sides by

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

\frac{6 sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )} = \frac{- 6 cos ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )}

Simplify

\frac{6 sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )} = - 6

Use the basic trigonometric identity:

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

6 tan (\frac{θ}{2}) = - 6

6 tan (\frac{θ}{2}) = - 6

Divide both sides by

6

6 tan (\frac{θ}{2}) = - 6

Divide both sides by

6

\frac{6 tan ( \frac{θ}{2} )}{6} = \frac{- 6}{6}

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{3 π}{4} + πn

\frac{θ}{2} = \frac{3 π}{4} + πn

Solve

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

\frac{θ}{2} = \frac{3 π}{4} + πn

Multiply both sides by

2

\frac{θ}{2} = \frac{3 π}{4} + πn

Multiply both sides by

2

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

Simplify

\frac{2 θ}{2} = 2 \cdot \frac{3 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{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

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

csc (θ) = - \frac{4}{3}

9 sin^{2} (θ) + 3 cos (θ) - 9 = - 2

2 cos (3 t) - 1 = 0

sin (x) + cos (x) - 1 = 0

tan (θ) = - 9

What is the general solution for 6sin(θ/2)=-6cos(θ/2) ?
The general solution for 6sin(θ/2)=-6cos(θ/2) is θ=(3pi)/2+2pin