What is the general solution for cos(3x+60)=0 ?

The general solution for cos(3x+60)=0 is x=(360n)/3+10,x=(360n)/3+70

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cos(3x+60)=0

Solution

cos (3 x + 6 0^{\circ}) = 0

Solution

x = \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}, x = \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

Radians

x = \frac{π}{18} + \frac{2 π}{3} n, x = \frac{7 π}{18} + \frac{2 π}{3} n

Solution steps

cos (3 x + 6 0^{\circ}) = 0

General solutions for

cos (3 x + 6 0^{\circ}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

3 x + 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n, 3 x + 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

3 x + 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n, 3 x + 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

Solve

3 x + 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}

3 x + 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

Move

6 0^{\circ}

to the right side

3 x + 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

Subtract

6 0^{\circ}

from both sides

3 x + 6 0^{\circ} - 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Simplify

3 x + 6 0^{\circ} - 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Simplify

3 x + 6 0^{\circ} - 6 0^{\circ} : 3 x

3 x + 6 0^{\circ} - 6 0^{\circ}

Add similar elements:

6 0^{\circ} - 6 0^{\circ} = 0

= 3 x

Simplify

9 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ} : 36 0^{\circ} n + 3 0^{\circ}

9 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Group like terms

= 36 0^{\circ} n + 9 0^{\circ} - 6 0^{\circ}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

9 0^{\circ} :

multiply the denominator and numerator by

3

9 0^{\circ} = \frac{18 0 ^{\circ} 3}{2 \cdot 3} = 9 0^{\circ}

For

6 0^{\circ} :

multiply the denominator and numerator by

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

= 9 0^{\circ} - 6 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{18 0 ^{\circ} 3 - 18 0 ^{\circ} 2}{6}

Add similar elements:

54 0^{\circ} - 36 0^{\circ} = 18 0^{\circ}

= 36 0^{\circ} n + 3 0^{\circ}

3 x = 36 0^{\circ} n + 3 0^{\circ}

3 x = 36 0^{\circ} n + 3 0^{\circ}

3 x = 36 0^{\circ} n + 3 0^{\circ}

Divide both sides by

3

3 x = 36 0^{\circ} n + 3 0^{\circ}

Divide both sides by

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{3 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{3 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{3 0 ^{\circ}}{3} : \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}

\frac{36 0 ^{\circ} n}{3} + \frac{3 0 ^{\circ}}{3}

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

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

Apply the fraction rule:

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

= \frac{18 0 ^{\circ}}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= 1 0^{\circ}

= \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}

Solve

3 x + 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n : x = \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

3 x + 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

Move

6 0^{\circ}

to the right side

3 x + 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n

Subtract

6 0^{\circ}

from both sides

3 x + 6 0^{\circ} - 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Simplify

3 x + 6 0^{\circ} - 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Simplify

3 x + 6 0^{\circ} - 6 0^{\circ} : 3 x

3 x + 6 0^{\circ} - 6 0^{\circ}

Add similar elements:

6 0^{\circ} - 6 0^{\circ} = 0

= 3 x

Simplify

27 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ} : 36 0^{\circ} n + 21 0^{\circ}

27 0^{\circ} + 36 0^{\circ} n - 6 0^{\circ}

Group like terms

= 36 0^{\circ} n - 6 0^{\circ} + 27 0^{\circ}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

6 0^{\circ} :

multiply the denominator and numerator by

2

6 0^{\circ} = \frac{18 0 ^{\circ} 2}{3 \cdot 2} = 6 0^{\circ}

For

27 0^{\circ} :

multiply the denominator and numerator by

3

27 0^{\circ} = \frac{54 0 ^{\circ} 3}{2 \cdot 3} = 27 0^{\circ}

= - 6 0^{\circ} + 27 0^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{- 18 0 ^{\circ} 2 + 162 0 ^{\circ}}{6}

Add similar elements:

- 36 0^{\circ} + 162 0^{\circ} = 126 0^{\circ}

= 36 0^{\circ} n + 21 0^{\circ}

3 x = 36 0^{\circ} n + 21 0^{\circ}

3 x = 36 0^{\circ} n + 21 0^{\circ}

3 x = 36 0^{\circ} n + 21 0^{\circ}

Divide both sides by

3

3 x = 36 0^{\circ} n + 21 0^{\circ}

Divide both sides by

3

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{21 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} = \frac{36 0 ^{\circ} n}{3} + \frac{21 0 ^{\circ}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{36 0 ^{\circ} n}{3} + \frac{21 0 ^{\circ}}{3} : \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

\frac{36 0 ^{\circ} n}{3} + \frac{21 0 ^{\circ}}{3}

\frac{21 0 ^{\circ}}{3} = 7 0^{\circ}

\frac{21 0 ^{\circ}}{3}

Apply the fraction rule:

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

= \frac{126 0 ^{\circ}}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= 7 0^{\circ}

= \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

x = \frac{36 0 ^{\circ} n}{3} + 1 0^{\circ}, x = \frac{36 0 ^{\circ} n}{3} + 7 0^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(3x+60)=0 ?
The general solution for cos(3x+60)=0 is x=(360n)/3+10,x=(360n)/3+70