What is the general solution for sin(x)=cos(40) ?

The general solution for sin(x)=cos(40) is x=360n+50,x=180-50+360n

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sin(x)=cos(40)

Solution

sin (x) = cos (4 0^{\circ})

Solution

x = 36 0^{\circ} n + 5 0^{\circ}, x = 18 0^{\circ} - 5 0^{\circ} + 36 0^{\circ} n

Radians

x = \frac{5 π}{18} + 2 πn, x = π - \frac{5 π}{18} + 2 πn

Solution steps

sin (x) = cos (4 0^{\circ})

Rewrite using trig identities

cos (4 0^{\circ})

Use the following identity:

cos (x) = sin (9 0^{\circ} - x)

sin (9 0^{\circ} - 4 0^{\circ})

sin (x) = sin (9 0^{\circ} - 4 0^{\circ})

Apply trig inverse properties

sin (x) = sin (9 0^{\circ} - 4 0^{\circ})

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

x = 9 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} - (9 0^{\circ} - 4 0^{\circ}) + 36 0^{\circ} n

x = 9 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} - (9 0^{\circ} - 4 0^{\circ}) + 36 0^{\circ} n

x = 9 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 5 0^{\circ}

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

Simplify

9 0^{\circ} - 4 0^{\circ} + 36 0^{\circ} n : 36 0^{\circ} n + 5 0^{\circ}

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

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} = 90 0^{\circ}

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

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

x = 18 0^{\circ} - (9 0^{\circ} - 4 0^{\circ}) + 36 0^{\circ} n : x = 18 0^{\circ} - 5 0^{\circ} + 36 0^{\circ} n

x = 18 0^{\circ} - (9 0^{\circ} - 4 0^{\circ}) + 36 0^{\circ} n

Join

9 0^{\circ} - 4 0^{\circ} : 5 0^{\circ}

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} = 90 0^{\circ}

= 5 0^{\circ}

x = 18 0^{\circ} - 5 0^{\circ} + 36 0^{\circ} n

x = 36 0^{\circ} n + 5 0^{\circ}, x = 18 0^{\circ} - 5 0^{\circ} + 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=cos(40) ?
The general solution for sin(x)=cos(40) is x=360n+50,x=180-50+360n