What is the general solution for sin(x+75)=(sqrt(3))/2 ?

The general solution for sin(x+75)=(sqrt(3))/2 is x=360n-15,x=360n+45

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sin(x+75)=(sqrt(3))/2

Solution

sin (x + 7 5^{\circ}) = \frac{3}{2}

Solution

x = 36 0^{\circ} n - 1 5^{\circ}, x = 36 0^{\circ} n + 4 5^{\circ}

Radians

x = - \frac{π}{12} + 2 πn, x = \frac{π}{4} + 2 πn

Solution steps

sin (x + 7 5^{\circ}) = \frac{3}{2}

General solutions for

sin (x + 7 5^{\circ}) = \frac{3}{2}

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 + 7 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x + 7 5^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

x + 7 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x + 7 5^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Solve

x + 7 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n - 1 5^{\circ}

x + 7 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Move

7 5^{\circ}

to the right side

x + 7 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Subtract

7 5^{\circ}

from both sides

x + 7 5^{\circ} - 7 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ}

Simplify

x + 7 5^{\circ} - 7 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ}

Simplify

x + 7 5^{\circ} - 7 5^{\circ} : x

x + 7 5^{\circ} - 7 5^{\circ}

Add similar elements:

7 5^{\circ} - 7 5^{\circ} = 0

= x

Simplify

6 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ} : 36 0^{\circ} n - 1 5^{\circ}

6 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ}

Group like terms

= 36 0^{\circ} n + 6 0^{\circ} - 7 5^{\circ}

Least Common Multiplier of

3, 12 : 12

3, 12

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

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

3

12

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 12

= 12

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

12

For

6 0^{\circ} :

multiply the denominator and numerator by

4

6 0^{\circ} = \frac{18 0 ^{\circ} 4}{3 \cdot 4} = 6 0^{\circ}

= 6 0^{\circ} - 7 5^{\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} 4 - 90 0 ^{\circ}}{12}

Add similar elements:

72 0^{\circ} - 90 0^{\circ} = - 18 0^{\circ}

= \frac{- 18 0 ^{\circ}}{12}

Apply the fraction rule:

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

= 36 0^{\circ} n - 1 5^{\circ}

x = 36 0^{\circ} n - 1 5^{\circ}

x = 36 0^{\circ} n - 1 5^{\circ}

x = 36 0^{\circ} n - 1 5^{\circ}

Solve

x + 7 5^{\circ} = 12 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 4 5^{\circ}

x + 7 5^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Move

7 5^{\circ}

to the right side

x + 7 5^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Subtract

7 5^{\circ}

from both sides

x + 7 5^{\circ} - 7 5^{\circ} = 12 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ}

Simplify

x + 7 5^{\circ} - 7 5^{\circ} = 12 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ}

Simplify

x + 7 5^{\circ} - 7 5^{\circ} : x

x + 7 5^{\circ} - 7 5^{\circ}

Add similar elements:

7 5^{\circ} - 7 5^{\circ} = 0

= x

Simplify

12 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ} : 36 0^{\circ} n + 4 5^{\circ}

12 0^{\circ} + 36 0^{\circ} n - 7 5^{\circ}

Group like terms

= 36 0^{\circ} n + 12 0^{\circ} - 7 5^{\circ}

Least Common Multiplier of

3, 12 : 12

3, 12

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

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

3

12

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 12

= 12

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

12

For

12 0^{\circ} :

multiply the denominator and numerator by

4

12 0^{\circ} = \frac{36 0 ^{\circ} 4}{3 \cdot 4} = 12 0^{\circ}

= 12 0^{\circ} - 7 5^{\circ}

Since the denominators are equal, combine the fractions:

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

= \frac{144 0 ^{\circ} - 90 0 ^{\circ}}{12}

Add similar elements:

144 0^{\circ} - 90 0^{\circ} = 54 0^{\circ}

= 4 5^{\circ}

Cancel the common factor:

3

= 36 0^{\circ} n + 4 5^{\circ}

x = 36 0^{\circ} n + 4 5^{\circ}

x = 36 0^{\circ} n + 4 5^{\circ}

x = 36 0^{\circ} n + 4 5^{\circ}

x = 36 0^{\circ} n - 1 5^{\circ}, x = 36 0^{\circ} n + 4 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x+75)=(sqrt(3))/2 ?
The general solution for sin(x+75)=(sqrt(3))/2 is x=360n-15,x=360n+45