What is the general solution for sin(θ-40)=1 ?

The general solution for sin(θ-40)=1 is θ=360n+130

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Popular Trigonometry >

sin(θ-40)=1

Solution

sin (θ - 4 0^{\circ}) = 1

Solution

θ = 36 0^{\circ} n + 13 0^{\circ}

Radians

θ = \frac{13 π}{18} + 2 πn

Solution steps

sin (θ - 4 0^{\circ}) = 1

General solutions for

sin (θ - 4 0^{\circ}) = 1

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}

θ - 4 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

θ - 4 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

Solve

θ - 4 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n : θ = 36 0^{\circ} n + 13 0^{\circ}

θ - 4 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

Move

4 0^{\circ}

to the right side

θ - 4 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n

Add

4 0^{\circ}

to both sides

θ - 4 0^{\circ} + 4 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n + 4 0^{\circ}

Simplify

θ - 4 0^{\circ} + 4 0^{\circ} = 9 0^{\circ} + 36 0^{\circ} n + 4 0^{\circ}

Simplify

θ - 4 0^{\circ} + 4 0^{\circ} : θ

θ - 4 0^{\circ} + 4 0^{\circ}

Add similar elements:

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

= θ

Simplify

9 0^{\circ} + 36 0^{\circ} n + 4 0^{\circ} : 36 0^{\circ} n + 13 0^{\circ}

9 0^{\circ} + 36 0^{\circ} n + 4 0^{\circ}

Group like terms

= 36 0^{\circ} n + 9 0^{\circ} + 4 0^{\circ}

Least Common Multiplier of

2, 9 : 18

2, 9

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

9

= 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 3 \cdot 3 = 18

= 18

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

18

For

9 0^{\circ} :

multiply the denominator and numerator by

9

9 0^{\circ} = \frac{18 0 ^{\circ} 9}{2 \cdot 9} = 9 0^{\circ}

For

4 0^{\circ} :

multiply the denominator and numerator by

2

4 0^{\circ} = \frac{36 0 ^{\circ} 2}{9 \cdot 2} = 4 0^{\circ}

= 9 0^{\circ} + 4 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} 9 + 72 0 ^{\circ}}{18}

Add similar elements:

162 0^{\circ} + 72 0^{\circ} = 234 0^{\circ}

= 36 0^{\circ} n + 13 0^{\circ}

θ = 36 0^{\circ} n + 13 0^{\circ}

θ = 36 0^{\circ} n + 13 0^{\circ}

θ = 36 0^{\circ} n + 13 0^{\circ}

θ = 36 0^{\circ} n + 13 0^{\circ}

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What is the general solution for sin(θ-40)=1 ?
The general solution for sin(θ-40)=1 is θ=360n+130