What is the general solution for 2cos(45-x)=1 ?

The general solution for 2cos(45-x)=1 is x=-360n-15,x=-360n-255

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2cos(45-x)=1

Solution

2 cos (4 5^{\circ} - x) = 1

Solution

x = - 36 0^{\circ} n - 1 5^{\circ}, x = - 36 0^{\circ} n - 25 5^{\circ}

Radians

x = - \frac{π}{12} - 2 πn, x = - \frac{17 π}{12} - 2 πn

Solution steps

2 cos (4 5^{\circ} - x) = 1

Divide both sides by

2

2 cos (4 5^{\circ} - x) = 1

Divide both sides by

2

\frac{2 cos ( 4 5 ^{\circ} - x )}{2} = \frac{1}{2}

Simplify

cos (4 5^{\circ} - x) = \frac{1}{2}

cos (4 5^{\circ} - x) = \frac{1}{2}

General solutions for

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

4 5^{\circ} - x = 6 0^{\circ} + 36 0^{\circ} n, 4 5^{\circ} - x = 30 0^{\circ} + 36 0^{\circ} n

4 5^{\circ} - x = 6 0^{\circ} + 36 0^{\circ} n, 4 5^{\circ} - x = 30 0^{\circ} + 36 0^{\circ} n

Solve

4 5^{\circ} - x = 6 0^{\circ} + 36 0^{\circ} n : x = - 36 0^{\circ} n - 1 5^{\circ}

4 5^{\circ} - x = 6 0^{\circ} + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} - x = 6 0^{\circ} + 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} - x - 4 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

4 5^{\circ} - x - 4 5^{\circ} = 6 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

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

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

Add similar elements:

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

= - x

Simplify

6 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ} : 36 0^{\circ} n + 1 5^{\circ}

6 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Group like terms

= 36 0^{\circ} n + 6 0^{\circ} - 4 5^{\circ}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 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

3

4

= 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}

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}

= 6 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} 4 - 18 0 ^{\circ} 3}{12}

Add similar elements:

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

= 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}

Divide both sides by

- 1

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

Divide both sides by

- 1

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

Simplify

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

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

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

\frac{36 0 ^{\circ} n}{- 1} + \frac{1 5 ^{\circ}}{- 1}

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

\frac{36 0 ^{\circ} n}{- 1}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= - 36 0^{\circ} n

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

\frac{1 5 ^{\circ}}{- 1} = - 1 5^{\circ}

\frac{1 5 ^{\circ}}{- 1}

Apply the fraction rule:

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

= - \frac{1 5 ^{\circ}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{1 5 ^{\circ}}{1} = 1 5^{\circ}

= - 1 5^{\circ}

= - 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

4 5^{\circ} - x = 30 0^{\circ} + 36 0^{\circ} n : x = - 36 0^{\circ} n - 25 5^{\circ}

4 5^{\circ} - x = 30 0^{\circ} + 36 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} - x = 30 0^{\circ} + 36 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} - x - 4 5^{\circ} = 30 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

4 5^{\circ} - x - 4 5^{\circ} = 30 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Simplify

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

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

Add similar elements:

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

= - x

Simplify

30 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ} : 36 0^{\circ} n + 25 5^{\circ}

30 0^{\circ} + 36 0^{\circ} n - 4 5^{\circ}

Group like terms

= 36 0^{\circ} n - 4 5^{\circ} + 30 0^{\circ}

Least Common Multiplier of

4, 3 : 12

4, 3

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

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

4

3

= 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

30 0^{\circ} :

multiply the denominator and numerator by

4

30 0^{\circ} = \frac{90 0 ^{\circ} 4}{3 \cdot 4} = 30 0^{\circ}

= - 4 5^{\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} 3 + 360 0 ^{\circ}}{12}

Add similar elements:

- 54 0^{\circ} + 360 0^{\circ} = 306 0^{\circ}

= 36 0^{\circ} n + 25 5^{\circ}

- x = 36 0^{\circ} n + 25 5^{\circ}

- x = 36 0^{\circ} n + 25 5^{\circ}

- x = 36 0^{\circ} n + 25 5^{\circ}

Divide both sides by

- 1

- x = 36 0^{\circ} n + 25 5^{\circ}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{36 0 ^{\circ} n}{- 1} + \frac{25 5 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{36 0 ^{\circ} n}{- 1} + \frac{25 5 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{36 0 ^{\circ} n}{- 1} + \frac{25 5 ^{\circ}}{- 1} : - 36 0^{\circ} n - 25 5^{\circ}

\frac{36 0 ^{\circ} n}{- 1} + \frac{25 5 ^{\circ}}{- 1}

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

\frac{36 0 ^{\circ} n}{- 1}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= - 36 0^{\circ} n

= - 36 0^{\circ} n + \frac{25 5 ^{\circ}}{- 1}

\frac{25 5 ^{\circ}}{- 1} = - 25 5^{\circ}

\frac{25 5 ^{\circ}}{- 1}

Apply the fraction rule:

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

= - \frac{25 5 ^{\circ}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{25 5 ^{\circ}}{1} = 25 5^{\circ}

= - 25 5^{\circ}

= - 36 0^{\circ} n - 25 5^{\circ}

x = - 36 0^{\circ} n - 25 5^{\circ}

x = - 36 0^{\circ} n - 25 5^{\circ}

x = - 36 0^{\circ} n - 25 5^{\circ}

x = - 36 0^{\circ} n - 1 5^{\circ}, x = - 36 0^{\circ} n - 25 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(45-x)=1 ?
The general solution for 2cos(45-x)=1 is x=-360n-15,x=-360n-255