What is the general solution for sin(30)=cos(x-20) ?

The general solution for sin(30)=cos(x-20) is x=360n+80,x=360n+320

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Popular Trigonometry >

sin(30)=cos(x-20)

Solution

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

Solution

x = 36 0^{\circ} n + 8 0^{\circ}, x = 36 0^{\circ} n + 32 0^{\circ}

Radians

x = \frac{4 π}{9} + 2 πn, x = \frac{16 π}{9} + 2 πn

Solution steps

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

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

sin (3 0^{\circ})

Use the following trivial identity:

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

sin (3 0^{\circ})

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}

= \frac{1}{2}

= \frac{1}{2}

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

Switch sides

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

General solutions for

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

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 - 2 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x - 2 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

x - 2 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x - 2 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Solve

x - 2 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 8 0^{\circ}

x - 2 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x - 2 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Add

2 0^{\circ}

to both sides

x - 2 0^{\circ} + 2 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

x - 2 0^{\circ} + 2 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

x - 2 0^{\circ} + 2 0^{\circ} : x

x - 2 0^{\circ} + 2 0^{\circ}

Add similar elements:

- 2 0^{\circ} + 2 0^{\circ} = 0

= x

Simplify

6 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ} : 36 0^{\circ} n + 8 0^{\circ}

6 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Group like terms

= 36 0^{\circ} n + 6 0^{\circ} + 2 0^{\circ}

Least Common Multiplier of

3, 9 : 9

3, 9

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

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

3

9

= 3 \cdot 3

Multiply the numbers:

3 \cdot 3 = 9

= 9

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

9

For

6 0^{\circ} :

multiply the denominator and numerator by

3

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

= 6 0^{\circ} + 2 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}}{9}

Add similar elements:

54 0^{\circ} + 18 0^{\circ} = 72 0^{\circ}

= 36 0^{\circ} n + 8 0^{\circ}

x = 36 0^{\circ} n + 8 0^{\circ}

x = 36 0^{\circ} n + 8 0^{\circ}

x = 36 0^{\circ} n + 8 0^{\circ}

Solve

x - 2 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 32 0^{\circ}

x - 2 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Move

2 0^{\circ}

to the right side

x - 2 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Add

2 0^{\circ}

to both sides

x - 2 0^{\circ} + 2 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

x - 2 0^{\circ} + 2 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Simplify

x - 2 0^{\circ} + 2 0^{\circ} : x

x - 2 0^{\circ} + 2 0^{\circ}

Add similar elements:

- 2 0^{\circ} + 2 0^{\circ} = 0

= x

Simplify

30 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ} : 36 0^{\circ} n + 32 0^{\circ}

30 0^{\circ} + 36 0^{\circ} n + 2 0^{\circ}

Group like terms

= 36 0^{\circ} n + 2 0^{\circ} + 30 0^{\circ}

Least Common Multiplier of

9, 3 : 9

9, 3

Least Common Multiplier (LCM)

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

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

9

3

= 3 \cdot 3

Multiply the numbers:

3 \cdot 3 = 9

= 9

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

9

For

30 0^{\circ} :

multiply the denominator and numerator by

3

30 0^{\circ} = \frac{90 0 ^{\circ} 3}{3 \cdot 3} = 30 0^{\circ}

= 2 0^{\circ} + 30 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} + 270 0 ^{\circ}}{9}

Add similar elements:

18 0^{\circ} + 270 0^{\circ} = 288 0^{\circ}

= 36 0^{\circ} n + 32 0^{\circ}

x = 36 0^{\circ} n + 32 0^{\circ}

x = 36 0^{\circ} n + 32 0^{\circ}

x = 36 0^{\circ} n + 32 0^{\circ}

x = 36 0^{\circ} n + 8 0^{\circ}, x = 36 0^{\circ} n + 32 0^{\circ}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(30)=cos(x-20) ?
The general solution for sin(30)=cos(x-20) is x=360n+80,x=360n+320