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Popular Trigonometry >

sin(x+16)=cos(11x+2)

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Solution

sin(x+16∘)=cos(11x+2)

Solution

x=108032400∘n+6660∘−180​,x=−9006660∘+32400∘n+180​
+1
Radians
x=216537π​​−61​+1080180π​n,x=−51​−180537π​​−900180π​n
Solution steps
sin(x+16∘)=cos(11x+2)
Rewrite using trig identities
sin(x+16∘)=cos(11x+2)
Use the following identity: cos(x)=sin(90∘−x)sin(x+16∘)=sin(90∘−(11x+2))
sin(x+16∘)=sin(90∘−(11x+2))
Apply trig inverse properties
sin(x+16∘)=sin(90∘−(11x+2))
sin(x)=sin(y)⇒x=y+2πn,x=π−y+2πnx+16∘=90∘−(11x+2)+360∘n,x+16∘=180∘−(90∘−(11x+2))+360∘n
x+16∘=90∘−(11x+2)+360∘n,x+16∘=180∘−(90∘−(11x+2))+360∘n
x+16∘=90∘−(11x+2)+360∘n:x=108032400∘n+6660∘−180​
x+16∘=90∘−(11x+2)+360∘n
Expand 90∘−(11x+2)+360∘n:90∘−11x−2+360∘n
90∘−(11x+2)+360∘n
−(11x+2):−11x−2
−(11x+2)
Distribute parentheses=−(11x)−(2)
Apply minus-plus rules+(−a)=−a=−11x−2
=90∘−11x−2+360∘n
x+16∘=90∘−11x−2+360∘n
Move 16∘to the right side
x+16∘=90∘−11x−2+360∘n
Subtract 16∘ from both sidesx+16∘−16∘=90∘−11x−2+360∘n−16∘
Simplify
x+16∘−16∘=90∘−11x−2+360∘n−16∘
Simplify x+16∘−16∘:x
x+16∘−16∘
Add similar elements: 16∘−16∘=0
=x
Simplify 90∘−11x−2+360∘n−16∘:−11x+360∘n+74∘−2
90∘−11x−2+360∘n−16∘
Group like terms=−11x+360∘n+90∘−16∘−2
Combine the fractions 90∘−16∘:74∘
90∘−16∘
Least Common Multiplier of 2,45:90
2,45
Least Common Multiplier (LCM)
Prime factorization of 2:2
2
2 is a prime number, therefore no factorization is possible=2
Prime factorization of 45:3⋅3⋅5
45
45divides by 345=15⋅3=3⋅15
15divides by 315=5⋅3=3⋅3⋅5
3,5 are all prime numbers, therefore no further factorization is possible=3⋅3⋅5
Multiply each factor the greatest number of times it occurs in either 2 or 45=2⋅3⋅3⋅5
Multiply the numbers: 2⋅3⋅3⋅5=90=90
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 90
For 90∘:multiply the denominator and numerator by 4590∘=2⋅45180∘45​=90∘
For 16∘:multiply the denominator and numerator by 216∘=45⋅2720∘2​=16∘
=90∘−16∘
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=90180∘45−1440∘​
Add similar elements: 8100∘−1440∘=6660∘=74∘
=−11x+360∘n+74∘−2
x=−11x+360∘n+74∘−2
x=−11x+360∘n+74∘−2
x=−11x+360∘n+74∘−2
Move 11xto the left side
x=−11x+360∘n+74∘−2
Add 11x to both sidesx+11x=−11x+360∘n+74∘−2+11x
Simplify12x=360∘n+74∘−2
12x=360∘n+74∘−2
Divide both sides by 12
12x=360∘n+74∘−2
Divide both sides by 121212x​=12360∘n​+1274∘​−122​
Simplify
1212x​=12360∘n​+1274∘​−122​
Simplify 1212x​:x
1212x​
Divide the numbers: 1212​=1=x
Simplify 12360∘n​+1274∘​−122​:108032400∘n+6660∘−180​
12360∘n​+1274∘​−122​
Apply rule ca​±cb​=ca±b​=12360∘n+74∘−2​
Join 360∘n+74∘−2:9032400∘n+6660∘−180​
360∘n+74∘−2
Convert element to fraction: 360∘n=90360∘n90​,2=902⋅90​=90360∘n⋅90​+74∘−902⋅90​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=90360∘n⋅90+6660∘−2⋅90​
Multiply the numbers: 2⋅90=180=9032400∘n+6660∘−180​
=129032400∘n+6660∘−180​​
Apply the fraction rule: acb​​=c⋅ab​=90⋅1232400∘n+6660∘−180​
Multiply the numbers: 90⋅12=1080=108032400∘n+6660∘−180​
x=108032400∘n+6660∘−180​
x=108032400∘n+6660∘−180​
x=108032400∘n+6660∘−180​
x+16∘=180∘−(90∘−(11x+2))+360∘n:x=−9006660∘+32400∘n+180​
x+16∘=180∘−(90∘−(11x+2))+360∘n
Expand 180∘−(90∘−(11x+2))+360∘n:180∘−90∘+11x+2+360∘n
180∘−(90∘−(11x+2))+360∘n
−(11x+2):−11x−2
−(11x+2)
Distribute parentheses=−(11x)−(2)
Apply minus-plus rules+(−a)=−a=−11x−2
=180∘−(−11x+90∘−2)+360∘n
−(90∘−11x−2):−90∘+11x+2
−(90∘−11x−2)
Distribute parentheses=−(90∘)−(−11x)−(−2)
Apply minus-plus rules−(−a)=a,−(a)=−a=−90∘+11x+2
=180∘−90∘+11x+2+360∘n
x+16∘=180∘−90∘+11x+2+360∘n
Move 16∘to the right side
x+16∘=180∘−90∘+11x+2+360∘n
Subtract 16∘ from both sidesx+16∘−16∘=180∘−90∘+11x+2+360∘n−16∘
Simplify
x+16∘−16∘=180∘−90∘+11x+2+360∘n−16∘
Simplify x+16∘−16∘:x
x+16∘−16∘
Add similar elements: 16∘−16∘=0
=x
Simplify 180∘−90∘+11x+2+360∘n−16∘:11x+180∘+360∘n−106∘+2
180∘−90∘+11x+2+360∘n−16∘
Group like terms=11x+180∘+360∘n−90∘−16∘+2
Combine the fractions −90∘−16∘:−106∘
−90∘−16∘
Least Common Multiplier of 2,45:90
2,45
Least Common Multiplier (LCM)
Prime factorization of 2:2
2
2 is a prime number, therefore no factorization is possible=2
Prime factorization of 45:3⋅3⋅5
45
45divides by 345=15⋅3=3⋅15
15divides by 315=5⋅3=3⋅3⋅5
3,5 are all prime numbers, therefore no further factorization is possible=3⋅3⋅5
Multiply each factor the greatest number of times it occurs in either 2 or 45=2⋅3⋅3⋅5
Multiply the numbers: 2⋅3⋅3⋅5=90=90
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 90
For 90∘:multiply the denominator and numerator by 4590∘=2⋅45180∘45​=90∘
For 16∘:multiply the denominator and numerator by 216∘=45⋅2720∘2​=16∘
=−90∘−16∘
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=90−180∘45−1440∘​
Add similar elements: −8100∘−1440∘=−9540∘=90−9540∘​
Apply the fraction rule: b−a​=−ba​=−106∘
=11x+180∘+360∘n−106∘+2
x=11x+180∘+360∘n−106∘+2
x=11x+180∘+360∘n−106∘+2
x=11x+180∘+360∘n−106∘+2
Move 11xto the left side
x=11x+180∘+360∘n−106∘+2
Subtract 11x from both sidesx−11x=11x+180∘+360∘n−106∘+2−11x
Simplify−10x=180∘+360∘n−106∘+2
−10x=180∘+360∘n−106∘+2
Divide both sides by −10
−10x=180∘+360∘n−106∘+2
Divide both sides by −10−10−10x​=−10180∘​+−10360∘n​−−10106∘​+−102​
Simplify
−10−10x​=−10180∘​+−10360∘n​−−10106∘​+−102​
Simplify −10−10x​:x
−10−10x​
Apply the fraction rule: −b−a​=ba​=1010x​
Divide the numbers: 1010​=1=x
Simplify −10180∘​+−10360∘n​−−10106∘​+−102​:−9006660∘+32400∘n+180​
−10180∘​+−10360∘n​−−10106∘​+−102​
Group like terms=−10180∘​+−102​+−10360∘n​−−10106∘​
Apply rule ca​±cb​=ca±b​=−10180∘+2+360∘n−106∘​
Apply the fraction rule: −ba​=−ba​=−10180∘+2+360∘n−106∘​
Join 180∘+2+360∘n−106∘:906660∘+32400∘n+180​
180∘+2+360∘n−106∘
Convert element to fraction: 180∘=180∘,2=902⋅90​,360∘n=90360∘n90​=180∘+902⋅90​+90360∘n⋅90​−106∘
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=90180∘90+2⋅90+360∘n⋅90−9540∘​
180∘90+2⋅90+360∘n⋅90−9540∘=6660∘+32400∘n+180
180∘90+2⋅90+360∘n⋅90−9540∘
Group like terms=16200∘−9540∘+2⋅16200∘n+2⋅90
Add similar elements: 16200∘−9540∘=6660∘=6660∘+2⋅16200∘n+2⋅90
Multiply the numbers: 2⋅90=180=6660∘+32400∘n+180
=906660∘+32400∘n+180​
=−10906660∘+32400∘n+180​​
Simplify 10906660∘+32400∘n+180​​:9006660∘+32400∘n+180​
10906660∘+32400∘n+180​​
Apply the fraction rule: acb​​=c⋅ab​=90⋅106660∘+32400∘n+180​
Multiply the numbers: 90⋅10=900=9006660∘+32400∘n+180​
=−9006660∘+32400∘n+180​
x=−9006660∘+32400∘n+180​
x=−9006660∘+32400∘n+180​
x=−9006660∘+32400∘n+180​
x=108032400∘n+6660∘−180​,x=−9006660∘+32400∘n+180​
x=108032400∘n+6660∘−180​,x=−9006660∘+32400∘n+180​

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