What is the general solution for 3tan(x+45)=sqrt(3) ?

The general solution for 3tan(x+45)=sqrt(3) is x=180n-15

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3tan(x+45)=sqrt(3)

Solution

3 tan (x + 4 5^{\circ}) = 3

Solution

x = 18 0^{\circ} n - 1 5^{\circ}

Radians

x = - \frac{π}{12} + πn

Solution steps

3 tan (x + 4 5^{\circ}) = 3

Divide both sides by

3

3 tan (x + 4 5^{\circ}) = 3

Divide both sides by

3

\frac{3 tan ( x + 4 5 ^{\circ} )}{3} = \frac{3}{3}

Simplify

tan (x + 4 5^{\circ}) = \frac{3}{3}

tan (x + 4 5^{\circ}) = \frac{3}{3}

General solutions for

tan (x + 4 5^{\circ}) = \frac{3}{3}

tan (x)

periodicity table with

18 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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x + 4 5^{\circ} = 3 0^{\circ} + 18 0^{\circ} n

x + 4 5^{\circ} = 3 0^{\circ} + 18 0^{\circ} n

Solve

x + 4 5^{\circ} = 3 0^{\circ} + 18 0^{\circ} n : x = 18 0^{\circ} n - 1 5^{\circ}

x + 4 5^{\circ} = 3 0^{\circ} + 18 0^{\circ} n

Move

4 5^{\circ}

to the right side

x + 4 5^{\circ} = 3 0^{\circ} + 18 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

x + 4 5^{\circ} - 4 5^{\circ} = 3 0^{\circ} + 18 0^{\circ} n - 4 5^{\circ}

Simplify

x + 4 5^{\circ} - 4 5^{\circ} = 3 0^{\circ} + 18 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} + 18 0^{\circ} n - 4 5^{\circ} : 18 0^{\circ} n - 1 5^{\circ}

3 0^{\circ} + 18 0^{\circ} n - 4 5^{\circ}

Group like terms

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

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

Apply the fraction rule:

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

= 18 0^{\circ} n - 1 5^{\circ}

x = 18 0^{\circ} n - 1 5^{\circ}

x = 18 0^{\circ} n - 1 5^{\circ}

x = 18 0^{\circ} n - 1 5^{\circ}

x = 18 0^{\circ} n - 1 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for 3tan(x+45)=sqrt(3) ?
The general solution for 3tan(x+45)=sqrt(3) is x=180n-15