Solution
73500=130000⋅sin(x)+0.15⋅130000⋅cos(x)
Solution
x=2.39936…+2πn,x=0.44444…+2πn
+1
Degrees
x=137.47362…∘+360∘n,x=25.46484…∘+360∘nSolution steps
73500=130000sin(x)+0.15⋅130000cos(x)
Subtract 0.15130000cos(x) from both sides130000sin(x)=73500−19500cos(x)
Square both sides(130000sin(x))2=(73500−19500cos(x))2
Subtract (73500−19500cos(x))2 from both sides1300002sin2(x)−735002+2866500000cos(x)−380250000cos2(x)=0
Rewrite using trig identities
−735002+1300002sin2(x)+2866500000cos(x)−380250000cos2(x)
Use the Pythagorean identity: cos2(x)+sin2(x)=1sin2(x)=1−cos2(x)=−735002+1300002(1−cos2(x))+2866500000cos(x)−380250000cos2(x)
−735002+(1−cos2(x))⋅1300002+2866500000cos(x)−380250000cos2(x)=0
Solve by substitution
−735002+(1−cos2(x))⋅1300002+2866500000cos(x)−380250000cos2(x)=0
Let: cos(x)=u−735002+(1−u2)⋅1300002+2866500000u−380250000u2=0
−735002+(1−u2)⋅1300002+2866500000u−380250000u2=0:u=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,u=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
−735002+(1−u2)⋅1300002+2866500000u−380250000u2=0
Expand −735002+(1−u2)⋅1300002+2866500000u−380250000u2:−735002+1300002−1300002u2+2866500000u−380250000u2
−735002+(1−u2)⋅1300002+2866500000u−380250000u2
=−735002+1300002(1−u2)+2866500000u−380250000u2
Expand 1300002(1−u2):1300002−1300002u2
1300002(1−u2)
Apply the distributive law: a(b−c)=ab−aca=1300002,b=1,c=u2=1300002⋅1−1300002u2
Multiply: 1300002⋅1=1300002=1300002−1300002u2
=−735002+1300002−1300002u2+2866500000u−380250000u2
−735002+1300002−1300002u2+2866500000u−380250000u2=0
Write in the standard form ax2+bx+c=0−(1300002+380250000)u2+2866500000u−735002+1300002=0
Solve with the quadratic formula
−(1300002+380250000)u2+2866500000u−735002+1300002=0
Quadratic Equation Formula:
For a=−1300002−380250000,b=2866500000,c=−735002+1300002u1,2=2(−1300002−380250000)−2866500000±28665000002−4(−1300002−380250000)(−735002+1300002)
u1,2=2(−1300002−380250000)−2866500000±28665000002−4(−1300002−380250000)(−735002+1300002)
28665000002−4(−1300002−380250000)(−735002+1300002)=28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
28665000002−4(−1300002−380250000)(−735002+1300002)
Expand 28665000002−4(−1300002−380250000)(−735002+1300002):28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
28665000002−4(−1300002−380250000)(−735002+1300002)
Expand −4(−1300002−380250000)(−735002+1300002):−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
Expand (−1300002−380250000)(−735002+1300002):514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
(−1300002−380250000)(−735002+1300002)
Apply FOIL method: (a+b)(c+d)=ac+ad+bc+bda=−1300002,b=−380250000,c=−735002,d=1300002=(−1300002)(−735002)+(−1300002)⋅1300002+(−380250000)(−735002)+(−380250000)⋅1300002
Apply minus-plus rules(−a)(−b)=ab,+(−a)=−a=1300002⋅735002−1300002⋅1300002+735002⋅380250000−1300002⋅380250000
Simplify 1300002⋅735002−1300002⋅1300002+735002⋅380250000−1300002⋅380250000:514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
1300002⋅735002−1300002⋅1300002+735002⋅380250000−1300002⋅380250000
1300002⋅735002=514⋅14958268416
1300002⋅735002
Factor integer 130000=24⋅54⋅13=(24⋅54⋅13)2⋅735002
Apply exponent rule: (ab)c=acbc(24⋅54⋅13)2=(24)2(54)2⋅132=(24)2(54)2⋅132⋅735002
Apply exponent rule: (ab)c=abc(24)2=24⋅2,(54)2=54⋅2=24⋅2⋅54⋅2⋅132⋅735002
Refine=28⋅58⋅132⋅735002
Factor integer 73500=53⋅22⋅147=28⋅58⋅132(22⋅53⋅147)2
Apply exponent rule: (ab)c=acbc(22⋅53⋅147)2=(22)2(53)2⋅1472=28⋅58⋅132(22)2(53)2⋅1472
Apply exponent rule: (ab)c=abc(22)2=22⋅2,(53)2=53⋅2=28⋅58⋅132⋅22⋅2⋅53⋅2⋅1472
Refine=28⋅58⋅132⋅24⋅56⋅1472
Apply exponent rule: ab⋅ac=ab+c28⋅24=28+4=58⋅132⋅28+4⋅56⋅1472
Add the numbers: 8+4=12=58⋅132⋅212⋅56⋅1472
Apply exponent rule: ab⋅ac=ab+c58⋅56=58+6=132⋅212⋅58+6⋅1472
Add the numbers: 8+6=14=132⋅212⋅514⋅1472
132=169=514⋅212⋅1472⋅169
212=4096=514⋅1472⋅169⋅4096
1472=21609=514⋅169⋅4096⋅21609
Multiply the numbers: 169⋅4096⋅21609=14958268416=514⋅14958268416
1300002⋅1300002=1300004
1300002⋅1300002
Apply exponent rule: ab⋅ac=ab+c1300002⋅1300002=1300002+2=1300002+2
Add the numbers: 2+2=4=1300004
=514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
=514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000
=−4(514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000)
Expand −4(514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000):−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
−4(514⋅14958268416−1300004+735002⋅380250000−1300002⋅380250000)
Distribute parentheses=(−4)⋅514⋅14958268416+(−4)(−1300004)+(−4)⋅735002⋅380250000+(−4)(−1300002⋅380250000)
Apply minus-plus rules+(−a)=−a,(−a)(−b)=ab=−514⋅4⋅14958268416+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000
Simplify −514⋅4⋅14958268416+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000:−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
−514⋅4⋅14958268416+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000
Multiply the numbers: 4⋅14958268416=59833073664=−514⋅59833073664+1300004⋅4−735002⋅4⋅380250000+1300002⋅4⋅380250000
Multiply the numbers: 4⋅380250000=1521000000=−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
=28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
u1,2=2(−1300002−380250000)−2866500000±28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
Separate the solutionsu1=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,u2=2(−1300002−380250000)−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
u=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
u=2(−1300002−380250000)−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000:−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
2(−1300002−380250000)−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
Apply the fraction rule: −b−a=ba−2866500000−28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000=−(1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
Apply the fraction rule: −ba=−ba=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
The solutions to the quadratic equation are:u=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,u=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
Substitute back u=cos(x)cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000,cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000:x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
Apply trig inverse properties
cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000
General solutions for cos(x)=2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000cos(x)=−a⇒x=arccos(−a)+2πn,x=−arccos(−a)+2πnx=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000:x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
Apply trig inverse properties
cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000
General solutions for cos(x)=−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000cos(x)=a⇒x=arccos(a)+2πn,x=2π−arccos(a)+2πnx=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
Combine all the solutionsx=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn,x=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
Verify solutions by plugging them into the original equation
Check the solutions by plugging them into 130000sin(x)+0.15130000cos(x)=73500
Remove the ones that don't agree with the equation.
Check the solution arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn:True
arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
Plug in n=1arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1
For 130000sin(x)+0.15130000cos(x)=73500plug inx=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1130000sin(arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)+0.15⋅130000cos(arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)=73500
Refine73500=73500
⇒True
Check the solution −arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn:False
−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn
Plug in n=1−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1
For 130000sin(x)+0.15130000cos(x)=73500plug inx=−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1130000sin(−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)+0.15⋅130000cos(−arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2π1)=73500
Refine−102241.68342…=73500
⇒False
Check the solution arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn:True
arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
Plug in n=1arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1
For 130000sin(x)+0.15130000cos(x)=73500plug inx=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1130000sin(arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)+0.15⋅130000cos(arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)=73500
Refine73500=73500
⇒True
Check the solution 2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn:False
2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
Plug in n=12π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1
For 130000sin(x)+0.15130000cos(x)=73500plug inx=2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1130000sin(2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)+0.15⋅130000cos(2π−arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2π1)=73500
Refine−38288.87892…=73500
⇒False
x=arccos(2(−1300002−380250000)−2866500000+28665000002−514⋅59833073664+1300004⋅4−735002⋅1521000000+1300002⋅1521000000)+2πn,x=arccos(−2(−1300002−380250000)1300004⋅4+28665000002+1300002⋅1521000000−514⋅59833073664−735002⋅1521000000+2866500000)+2πn
Show solutions in decimal formx=2.39936…+2πn,x=0.44444…+2πn
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Frequently Asked Questions (FAQ)
What is the general solution for 73500=130000*sin(x)+0.15*130000*cos(x) ?
The general solution for 73500=130000*sin(x)+0.15*130000*cos(x) is x=2.39936…+2pin,x=0.44444…+2pin