What is the general solution for 7sin(θ/2)=-7cos(θ/2) ?

The general solution for 7sin(θ/2)=-7cos(θ/2) is θ=(3pi)/2+2pin

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Popular Trigonometry >

7sin(θ/2)=-7cos(θ/2)

Solution

7 sin (\frac{θ}{2}) = - 7 cos (\frac{θ}{2})

Solution

θ = \frac{3 π}{2} + 2 πn

Degrees

θ = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

7 sin (\frac{θ}{2}) = - 7 cos (\frac{θ}{2})

Rewrite using trig identities

7 sin (\frac{θ}{2}) = - 7 cos (\frac{θ}{2})

Divide both sides by

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

\frac{7 sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )} = \frac{- 7 cos ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )}

Simplify

\frac{7 sin ( \frac{θ}{2} )}{cos ( \frac{θ}{2} )} = - 7

Use the basic trigonometric identity:

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

7 tan (\frac{θ}{2}) = - 7

7 tan (\frac{θ}{2}) = - 7

Divide both sides by

7

7 tan (\frac{θ}{2}) = - 7

Divide both sides by

7

\frac{7 tan ( \frac{θ}{2} )}{7} = \frac{- 7}{7}

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 7sin(θ/2)=-7cos(θ/2) ?
The general solution for 7sin(θ/2)=-7cos(θ/2) is θ=(3pi)/2+2pin