What is the general solution for sin(x)cos(x-60)=0 ?

The general solution for sin(x)cos(x-60)=0 is x=360n,x=180+360n,x=360n+150,x=360n+330

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Popular Trigonometry >

sin(x)cos(x-60)=0

Solution

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

Solution

x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 15 0^{\circ}, x = 36 0^{\circ} n + 33 0^{\circ}

Radians

x = 0 + 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

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

Solving each part separately

sin (x) = 0 or cos (x - 6 0^{\circ}) = 0

sin (x) = 0 : x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

sin (x) = 0

General solutions for

sin (x) = 0

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

x = 0 + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

x = 0 + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solve

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

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

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

x = 36 0^{\circ} n

x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

cos (x - 6 0^{\circ}) = 0 : x = 36 0^{\circ} n + 15 0^{\circ}, x = 36 0^{\circ} n + 33 0^{\circ}

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

General solutions for

cos (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}

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

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

Solve

x - 6 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 15 0^{\circ}

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

Move

6 0^{\circ}

to the right side

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

Add

6 0^{\circ}

to both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

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

= x

Simplify

9 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 15 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} = 90 0^{\circ}

= 36 0^{\circ} n + 15 0^{\circ}

x = 36 0^{\circ} n + 15 0^{\circ}

x = 36 0^{\circ} n + 15 0^{\circ}

x = 36 0^{\circ} n + 15 0^{\circ}

Solve

x - 6 0^{\circ} = 27 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 33 0^{\circ}

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

Move

6 0^{\circ}

to the right side

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

Add

6 0^{\circ}

to both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

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

= x

Simplify

27 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 33 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} = 198 0^{\circ}

= 36 0^{\circ} n + 33 0^{\circ}

x = 36 0^{\circ} n + 33 0^{\circ}

x = 36 0^{\circ} n + 33 0^{\circ}

x = 36 0^{\circ} n + 33 0^{\circ}

x = 36 0^{\circ} n + 15 0^{\circ}, x = 36 0^{\circ} n + 33 0^{\circ}

Combine all the solutions

x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 15 0^{\circ}, x = 36 0^{\circ} n + 33 0^{\circ}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)cos(x-60)=0 ?
The general solution for sin(x)cos(x-60)=0 is x=360n,x=180+360n,x=360n+150,x=360n+330