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Popular Trigonometry >

15sin(x)+6cos(x)-3=0

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Solution

15sin(x)+6cos(x)−3=0

Solution

x=π−0.56728…+2πn,x=−0.19372…+2πn
+1
Degrees
x=147.49691…∘+360∘n,x=−11.09973…∘+360∘n
Solution steps
15sin(x)+6cos(x)−3=0
Subtract 6cos(x) from both sides15sin(x)−3=−6cos(x)
Square both sides(15sin(x)−3)2=(−6cos(x))2
Subtract (−6cos(x))2 from both sides(15sin(x)−3)2−36cos2(x)=0
Rewrite using trig identities
(−3+15sin(x))2−36cos2(x)
Use the Pythagorean identity: cos2(x)+sin2(x)=1cos2(x)=1−sin2(x)=(−3+15sin(x))2−36(1−sin2(x))
Simplify (−3+15sin(x))2−36(1−sin2(x)):261sin2(x)−90sin(x)−27
(−3+15sin(x))2−36(1−sin2(x))
(−3+15sin(x))2:9−90sin(x)+225sin2(x)
Apply Perfect Square Formula: (a+b)2=a2+2ab+b2a=−3,b=15sin(x)
=(−3)2+2(−3)⋅15sin(x)+(15sin(x))2
Simplify (−3)2+2(−3)⋅15sin(x)+(15sin(x))2:9−90sin(x)+225sin2(x)
(−3)2+2(−3)⋅15sin(x)+(15sin(x))2
Remove parentheses: (−a)=−a=(−3)2−2⋅3⋅15sin(x)+(15sin(x))2
(−3)2=9
(−3)2
Apply exponent rule: (−a)n=an,if n is even(−3)2=32=32
32=9=9
2⋅3⋅15sin(x)=90sin(x)
2⋅3⋅15sin(x)
Multiply the numbers: 2⋅3⋅15=90=90sin(x)
(15sin(x))2=225sin2(x)
(15sin(x))2
Apply exponent rule: (a⋅b)n=anbn=152sin2(x)
152=225=225sin2(x)
=9−90sin(x)+225sin2(x)
=9−90sin(x)+225sin2(x)
=9−90sin(x)+225sin2(x)−36(1−sin2(x))
Expand −36(1−sin2(x)):−36+36sin2(x)
−36(1−sin2(x))
Apply the distributive law: a(b−c)=ab−aca=−36,b=1,c=sin2(x)=−36⋅1−(−36)sin2(x)
Apply minus-plus rules−(−a)=a=−36⋅1+36sin2(x)
Multiply the numbers: 36⋅1=36=−36+36sin2(x)
=9−90sin(x)+225sin2(x)−36+36sin2(x)
Simplify 9−90sin(x)+225sin2(x)−36+36sin2(x):261sin2(x)−90sin(x)−27
9−90sin(x)+225sin2(x)−36+36sin2(x)
Group like terms=−90sin(x)+225sin2(x)+36sin2(x)+9−36
Add similar elements: 225sin2(x)+36sin2(x)=261sin2(x)=−90sin(x)+261sin2(x)+9−36
Add/Subtract the numbers: 9−36=−27=261sin2(x)−90sin(x)−27
=261sin2(x)−90sin(x)−27
=261sin2(x)−90sin(x)−27
−27+261sin2(x)−90sin(x)=0
Solve by substitution
−27+261sin2(x)−90sin(x)=0
Let: sin(x)=u−27+261u2−90u=0
−27+261u2−90u=0:u=295+47​​,u=295−47​​
−27+261u2−90u=0
Write in the standard form ax2+bx+c=0261u2−90u−27=0
Solve with the quadratic formula
261u2−90u−27=0
Quadratic Equation Formula:
For a=261,b=−90,c=−27u1,2​=2⋅261−(−90)±(−90)2−4⋅261(−27)​​
u1,2​=2⋅261−(−90)±(−90)2−4⋅261(−27)​​
(−90)2−4⋅261(−27)​=727​
(−90)2−4⋅261(−27)​
Apply rule −(−a)=a=(−90)2+4⋅261⋅27​
Apply exponent rule: (−a)n=an,if n is even(−90)2=902=902+4⋅261⋅27​
Multiply the numbers: 4⋅261⋅27=28188=902+28188​
902=8100=8100+28188​
Add the numbers: 8100+28188=36288=36288​
Prime factorization of 36288:26⋅34⋅7
36288
36288divides by 236288=18144⋅2=2⋅18144
18144divides by 218144=9072⋅2=2⋅2⋅9072
9072divides by 29072=4536⋅2=2⋅2⋅2⋅4536
4536divides by 24536=2268⋅2=2⋅2⋅2⋅2⋅2268
2268divides by 22268=1134⋅2=2⋅2⋅2⋅2⋅2⋅1134
1134divides by 21134=567⋅2=2⋅2⋅2⋅2⋅2⋅2⋅567
567divides by 3567=189⋅3=2⋅2⋅2⋅2⋅2⋅2⋅3⋅189
189divides by 3189=63⋅3=2⋅2⋅2⋅2⋅2⋅2⋅3⋅3⋅63
63divides by 363=21⋅3=2⋅2⋅2⋅2⋅2⋅2⋅3⋅3⋅3⋅21
21divides by 321=7⋅3=2⋅2⋅2⋅2⋅2⋅2⋅3⋅3⋅3⋅3⋅7
2,3,7 are all prime numbers, therefore no further factorization is possible=2⋅2⋅2⋅2⋅2⋅2⋅3⋅3⋅3⋅3⋅7
=26⋅34⋅7
=26⋅34⋅7​
Apply radical rule: =7​26​34​
Apply radical rule: 26​=226​=23=237​34​
Apply radical rule: 34​=324​=32=23⋅327​
Refine=727​
u1,2​=2⋅261−(−90)±727​​
Separate the solutionsu1​=2⋅261−(−90)+727​​,u2​=2⋅261−(−90)−727​​
u=2⋅261−(−90)+727​​:295+47​​
2⋅261−(−90)+727​​
Apply rule −(−a)=a=2⋅26190+727​​
Multiply the numbers: 2⋅261=522=52290+727​​
Factor 90+727​:18(5+47​)
90+727​
Rewrite as=18⋅5+18⋅47​
Factor out common term 18=18(5+47​)
=52218(5+47​)​
Cancel the common factor: 18=295+47​​
u=2⋅261−(−90)−727​​:295−47​​
2⋅261−(−90)−727​​
Apply rule −(−a)=a=2⋅26190−727​​
Multiply the numbers: 2⋅261=522=52290−727​​
Factor 90−727​:18(5−47​)
90−727​
Rewrite as=18⋅5−18⋅47​
Factor out common term 18=18(5−47​)
=52218(5−47​)​
Cancel the common factor: 18=295−47​​
The solutions to the quadratic equation are:u=295+47​​,u=295−47​​
Substitute back u=sin(x)sin(x)=295+47​​,sin(x)=295−47​​
sin(x)=295+47​​,sin(x)=295−47​​
sin(x)=295+47​​:x=arcsin(295+47​​)+2πn,x=π−arcsin(295+47​​)+2πn
sin(x)=295+47​​
Apply trig inverse properties
sin(x)=295+47​​
General solutions for sin(x)=295+47​​sin(x)=a⇒x=arcsin(a)+2πn,x=π−arcsin(a)+2πnx=arcsin(295+47​​)+2πn,x=π−arcsin(295+47​​)+2πn
x=arcsin(295+47​​)+2πn,x=π−arcsin(295+47​​)+2πn
sin(x)=295−47​​:x=arcsin(295−47​​)+2πn,x=π+arcsin(−295−47​​)+2πn
sin(x)=295−47​​
Apply trig inverse properties
sin(x)=295−47​​
General solutions for sin(x)=295−47​​sin(x)=−a⇒x=arcsin(−a)+2πn,x=π+arcsin(a)+2πnx=arcsin(295−47​​)+2πn,x=π+arcsin(−295−47​​)+2πn
x=arcsin(295−47​​)+2πn,x=π+arcsin(−295−47​​)+2πn
Combine all the solutionsx=arcsin(295+47​​)+2πn,x=π−arcsin(295+47​​)+2πn,x=arcsin(295−47​​)+2πn,x=π+arcsin(−295−47​​)+2πn
Verify solutions by plugging them into the original equation
Check the solutions by plugging them into 15sin(x)+6cos(x)−3=0
Remove the ones that don't agree with the equation.
Check the solution arcsin(295+47​​)+2πn:False
arcsin(295+47​​)+2πn
Plug in n=1arcsin(295+47​​)+2π1
For 15sin(x)+6cos(x)−3=0plug inx=arcsin(295+47​​)+2π115sin(arcsin(295+47​​)+2π1)+6cos(arcsin(295+47​​)+2π1)−3=0
Refine10.12035…=0
⇒False
Check the solution π−arcsin(295+47​​)+2πn:True
π−arcsin(295+47​​)+2πn
Plug in n=1π−arcsin(295+47​​)+2π1
For 15sin(x)+6cos(x)−3=0plug inx=π−arcsin(295+47​​)+2π115sin(π−arcsin(295+47​​)+2π1)+6cos(π−arcsin(295+47​​)+2π1)−3=0
Refine0=0
⇒True
Check the solution arcsin(295−47​​)+2πn:True
arcsin(295−47​​)+2πn
Plug in n=1arcsin(295−47​​)+2π1
For 15sin(x)+6cos(x)−3=0plug inx=arcsin(295−47​​)+2π115sin(arcsin(295−47​​)+2π1)+6cos(arcsin(295−47​​)+2π1)−3=0
Refine0=0
⇒True
Check the solution π+arcsin(−295−47​​)+2πn:False
π+arcsin(−295−47​​)+2πn
Plug in n=1π+arcsin(−295−47​​)+2π1
For 15sin(x)+6cos(x)−3=0plug inx=π+arcsin(−295−47​​)+2π115sin(π+arcsin(−295−47​​)+2π1)+6cos(π+arcsin(−295−47​​)+2π1)−3=0
Refine−11.77552…=0
⇒False
x=π−arcsin(295+47​​)+2πn,x=arcsin(295−47​​)+2πn
Show solutions in decimal formx=π−0.56728…+2πn,x=−0.19372…+2πn

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Frequently Asked Questions (FAQ)

  • What is the general solution for 15sin(x)+6cos(x)-3=0 ?

    The general solution for 15sin(x)+6cos(x)-3=0 is x=pi-0.56728…+2pin,x=-0.19372…+2pin
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