What is the general solution for csc(x-45)=2 ?

The general solution for csc(x-45)=2 is x=360n+75,x=360n+195

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

csc(x-45)=2

Solution

csc (x - 4 5^{\circ}) = 2

Solution

x = 36 0^{\circ} n + 7 5^{\circ}, x = 36 0^{\circ} n + 19 5^{\circ}

Radians

x = \frac{5 π}{12} + 2 πn, x = \frac{13 π}{12} + 2 πn

Solution steps

csc (x - 4 5^{\circ}) = 2

General solutions for

csc (x - 4 5^{\circ}) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x - 4 5^{\circ} = 3 0^{\circ} + 36 0^{\circ} n, x - 4 5^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

x - 4 5^{\circ} = 3 0^{\circ} + 36 0^{\circ} n, x - 4 5^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

Solve

x - 4 5^{\circ} = 3 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 7 5^{\circ}

x - 4 5^{\circ} = 3 0^{\circ} + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

x - 4 5^{\circ} = 3 0^{\circ} + 36 0^{\circ} n

Add

4 5^{\circ}

to both sides

x - 4 5^{\circ} + 4 5^{\circ} = 3 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ}

Simplify

x - 4 5^{\circ} + 4 5^{\circ} = 3 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ}

Simplify

x - 4 5^{\circ} + 4 5^{\circ} : x

x - 4 5^{\circ} + 4 5^{\circ}

Add similar elements:

- 4 5^{\circ} + 4 5^{\circ} = 0

= x

Simplify

3 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ} : 36 0^{\circ} n + 7 5^{\circ}

3 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ}

Group like terms

= 36 0^{\circ} n + 3 0^{\circ} + 4 5^{\circ}

Least Common Multiplier of

6, 4 : 12

6, 4

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 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

3 0^{\circ} :

multiply the denominator and numerator by

2

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

For

4 5^{\circ} :

multiply the denominator and numerator by

3

4 5^{\circ} = \frac{18 0 ^{\circ} 3}{4 \cdot 3} = 4 5^{\circ}

= 3 0^{\circ} + 4 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} 2 + 18 0 ^{\circ} 3}{12}

Add similar elements:

36 0^{\circ} + 54 0^{\circ} = 90 0^{\circ}

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

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

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

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

Solve

x - 4 5^{\circ} = 15 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 19 5^{\circ}

x - 4 5^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

x - 4 5^{\circ} = 15 0^{\circ} + 36 0^{\circ} n

Add

4 5^{\circ}

to both sides

x - 4 5^{\circ} + 4 5^{\circ} = 15 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ}

Simplify

x - 4 5^{\circ} + 4 5^{\circ} = 15 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ}

Simplify

x - 4 5^{\circ} + 4 5^{\circ} : x

x - 4 5^{\circ} + 4 5^{\circ}

Add similar elements:

- 4 5^{\circ} + 4 5^{\circ} = 0

= x

Simplify

15 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ} : 36 0^{\circ} n + 19 5^{\circ}

15 0^{\circ} + 36 0^{\circ} n + 4 5^{\circ}

Group like terms

= 36 0^{\circ} n + 4 5^{\circ} + 15 0^{\circ}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

4

6

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 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

4 5^{\circ} :

multiply the denominator and numerator by

3

4 5^{\circ} = \frac{18 0 ^{\circ} 3}{4 \cdot 3} = 4 5^{\circ}

For

15 0^{\circ} :

multiply the denominator and numerator by

2

15 0^{\circ} = \frac{90 0 ^{\circ} 2}{6 \cdot 2} = 15 0^{\circ}

= 4 5^{\circ} + 15 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 + 180 0 ^{\circ}}{12}

Add similar elements:

54 0^{\circ} + 180 0^{\circ} = 234 0^{\circ}

= 36 0^{\circ} n + 19 5^{\circ}

x = 36 0^{\circ} n + 19 5^{\circ}

x = 36 0^{\circ} n + 19 5^{\circ}

x = 36 0^{\circ} n + 19 5^{\circ}

x = 36 0^{\circ} n + 7 5^{\circ}, x = 36 0^{\circ} n + 19 5^{\circ}

Graph

Popular Examples

-5sin(2t)=0

(cos(x)-1)=0

1/2 cos^2(2x)+tan(162)=0

1+cot^2(a)=tan^2(a)

tan(α)=(12)/(6sqrt(3))

Frequently Asked Questions (FAQ)

What is the general solution for csc(x-45)=2 ?
The general solution for csc(x-45)=2 is x=360n+75,x=360n+195