What is the general solution for 6sin(2x-30)+2=0 ?

The general solution for 6sin(2x-30)+2=0 is x=180n+15-(0.33983…)/2 ,x=180n+90+15+(0.33983…)/2

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Popular Trigonometry >

6sin(2x-30)+2=0

Solution

6 sin (2 x - 3 0^{\circ}) + 2 = 0

Solution

x = 18 0^{\circ} n + 1 5^{\circ} - \frac{0.33983 \dots}{2}, x = 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{0.33983 \dots}{2}

Radians

x = \frac{π}{12} - \frac{0.33983 \dots}{2} + πn, x = \frac{π}{2} + \frac{π}{12} + \frac{0.33983 \dots}{2} + πn

Solution steps

6 sin (2 x - 3 0^{\circ}) + 2 = 0

Move

2

to the right side

6 sin (2 x - 3 0^{\circ}) + 2 = 0

Subtract

2

from both sides

6 sin (2 x - 3 0^{\circ}) + 2 - 2 = 0 - 2

Simplify

6 sin (2 x - 3 0^{\circ}) = - 2

6 sin (2 x - 3 0^{\circ}) = - 2

Divide both sides by

6

6 sin (2 x - 3 0^{\circ}) = - 2

Divide both sides by

6

\frac{6 sin ( 2 x - 3 0 ^{\circ} )}{6} = \frac{- 2}{6}

Simplify

\frac{6 sin ( 2 x - 3 0 ^{\circ} )}{6} = \frac{- 2}{6}

Simplify

\frac{6 sin ( 2 x - 3 0 ^{\circ} )}{6} : sin (2 x - 3 0^{\circ})

\frac{6 sin ( 2 x - 3 0 ^{\circ} )}{6}

Divide the numbers:

\frac{6}{6} = 1

= sin (2 x - 3 0^{\circ})

Simplify

\frac{- 2}{6} : - \frac{1}{3}

\frac{- 2}{6}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{6}

Cancel the common factor:

2

= - \frac{1}{3}

sin (2 x - 3 0^{\circ}) = - \frac{1}{3}

sin (2 x - 3 0^{\circ}) = - \frac{1}{3}

sin (2 x - 3 0^{\circ}) = - \frac{1}{3}

Apply trig inverse properties

sin (2 x - 3 0^{\circ}) = - \frac{1}{3}

General solutions for

sin (2 x - 3 0^{\circ}) = - \frac{1}{3}

sin (x) = - a \Rightarrow x = arcsin (- a) + 36 0^{\circ} n, x = 18 0^{\circ} + arcsin (a) + 36 0^{\circ} n

2 x - 3 0^{\circ} = arcsin (- \frac{1}{3}) + 36 0^{\circ} n, 2 x - 3 0^{\circ} = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n

2 x - 3 0^{\circ} = arcsin (- \frac{1}{3}) + 36 0^{\circ} n, 2 x - 3 0^{\circ} = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n

Solve

2 x - 3 0^{\circ} = arcsin (- \frac{1}{3}) + 36 0^{\circ} n : x = 18 0^{\circ} n + 1 5^{\circ} - \frac{arcsin ( \frac{1}{3} )}{2}

2 x - 3 0^{\circ} = arcsin (- \frac{1}{3}) + 36 0^{\circ} n

Simplify

arcsin (- \frac{1}{3}) + 36 0^{\circ} n : - arcsin (\frac{1}{3}) + 36 0^{\circ} n

arcsin (- \frac{1}{3}) + 36 0^{\circ} n

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - arcsin (\frac{1}{3}) + 36 0^{\circ} n

2 x - 3 0^{\circ} = - arcsin (\frac{1}{3}) + 36 0^{\circ} n

Move

3 0^{\circ}

to the right side

2 x - 3 0^{\circ} = - arcsin (\frac{1}{3}) + 36 0^{\circ} n

Add

3 0^{\circ}

to both sides

2 x - 3 0^{\circ} + 3 0^{\circ} = - arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

Simplify

2 x = - arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

2 x = - arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

Divide both sides by

2

2 x = - arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = - \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2} : 18 0^{\circ} n + 1 5^{\circ} - \frac{arcsin ( \frac{1}{3} )}{2}

- \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2}

Group like terms

= \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2} - \frac{arcsin ( \frac{1}{3} )}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{3 0 ^{\circ}}{2} = 1 5^{\circ}

\frac{3 0 ^{\circ}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{18 0 ^{\circ}}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= 1 5^{\circ}

= 18 0^{\circ} n + 1 5^{\circ} - \frac{arcsin ( \frac{1}{3} )}{2}

x = 18 0^{\circ} n + 1 5^{\circ} - \frac{arcsin ( \frac{1}{3} )}{2}

x = 18 0^{\circ} n + 1 5^{\circ} - \frac{arcsin ( \frac{1}{3} )}{2}

x = 18 0^{\circ} n + 1 5^{\circ} - \frac{arcsin ( \frac{1}{3} )}{2}

Solve

2 x - 3 0^{\circ} = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n : x = 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

2 x - 3 0^{\circ} = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n

Move

3 0^{\circ}

to the right side

2 x - 3 0^{\circ} = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n

Add

3 0^{\circ}

to both sides

2 x - 3 0^{\circ} + 3 0^{\circ} = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

Simplify

2 x = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

2 x = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

Divide both sides by

2

2 x = 18 0^{\circ} + arcsin (\frac{1}{3}) + 36 0^{\circ} n + 3 0^{\circ}

Divide both sides by

2

\frac{2 x}{2} = 9 0^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} = 9 0^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

9 0^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2} : 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

9 0^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2}

Group like terms

= 9 0^{\circ} + \frac{36 0 ^{\circ} n}{2} + \frac{3 0 ^{\circ}}{2} + \frac{arcsin ( \frac{1}{3} )}{2}

\frac{36 0 ^{\circ} n}{2} = 18 0^{\circ} n

\frac{36 0 ^{\circ} n}{2}

Divide the numbers:

\frac{2}{2} = 1

= 18 0^{\circ} n

\frac{3 0 ^{\circ}}{2} = 1 5^{\circ}

\frac{3 0 ^{\circ}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{18 0 ^{\circ}}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= 1 5^{\circ}

= 9 0^{\circ} + 18 0^{\circ} n + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

Group like terms

= 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

x = 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

x = 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

x = 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

x = 18 0^{\circ} n + 1 5^{\circ} - \frac{arcsin ( \frac{1}{3} )}{2}, x = 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{arcsin ( \frac{1}{3} )}{2}

Show solutions in decimal form

x = 18 0^{\circ} n + 1 5^{\circ} - \frac{0.33983 \dots}{2}, x = 18 0^{\circ} n + 9 0^{\circ} + 1 5^{\circ} + \frac{0.33983 \dots}{2}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6sin(2x-30)+2=0 ?
The general solution for 6sin(2x-30)+2=0 is x=180n+15-(0.33983…)/2 ,x=180n+90+15+(0.33983…)/2