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Popular Trigonometry >

sin(x)=cos(40)

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Solution

sin(x)=cos(40∘)

Solution

x=360∘n+50∘,x=180∘−50∘+360∘n
+1
Radians
x=185π​+2πn,x=π−185π​+2πn
Solution steps
sin(x)=cos(40∘)
Rewrite using trig identities
cos(40∘)
Use the following identity: cos(x)=sin(90∘−x)sin(90∘−40∘)
sin(x)=sin(90∘−40∘)
Apply trig inverse properties
sin(x)=sin(90∘−40∘)
sin(x)=sin(y)⇒x=y+2πn,x=π−y+2πnx=90∘−40∘+360∘n,x=180∘−(90∘−40∘)+360∘n
x=90∘−40∘+360∘n,x=180∘−(90∘−40∘)+360∘n
x=90∘−40∘+360∘n:x=360∘n+50∘
x=90∘−40∘+360∘n
Simplify 90∘−40∘+360∘n:360∘n+50∘
90∘−40∘+360∘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⋅3
9
9divides by 39=3⋅3=3⋅3
Multiply each factor the greatest number of times it occurs in either 2 or 9=2⋅3⋅3
Multiply the numbers: 2⋅3⋅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 90∘:multiply the denominator and numerator by 990∘=2⋅9180∘9​=90∘
For 40∘:multiply the denominator and numerator by 240∘=9⋅2360∘2​=40∘
=90∘−40∘
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=18180∘9−720∘​
Add similar elements: 1620∘−720∘=900∘=360∘n+50∘
x=360∘n+50∘
x=180∘−(90∘−40∘)+360∘n:x=180∘−50∘+360∘n
x=180∘−(90∘−40∘)+360∘n
Join 90∘−40∘:50∘
90∘−40∘
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⋅3
9
9divides by 39=3⋅3=3⋅3
Multiply each factor the greatest number of times it occurs in either 2 or 9=2⋅3⋅3
Multiply the numbers: 2⋅3⋅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 90∘:multiply the denominator and numerator by 990∘=2⋅9180∘9​=90∘
For 40∘:multiply the denominator and numerator by 240∘=9⋅2360∘2​=40∘
=90∘−40∘
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=18180∘9−720∘​
Add similar elements: 1620∘−720∘=900∘=50∘
x=180∘−50∘+360∘n
x=360∘n+50∘,x=180∘−50∘+360∘n

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Popular Examples

5cos(x)+sqrt(2)=3cos(x),0<= x<= 2pi2tan(2x+10)=-5.5sin(t)=sin(2t)(sqrt(3))/3 =-1/3*tan(θ)tan(x+pi/3)=sqrt(3)

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
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