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Popular Trigonometry >

16sin(3x)cos(x)+8sqrt(3)sin(3x)-2cos(x)-sqrt(3)=0

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Solution

16sin(3x)cos(x)+83​sin(3x)−2cos(x)−3​=0

Solution

x=30.12532…​+32πn​,x=3π​−30.12532…​+32πn​,x=65π​+2πn,x=67π​+2πn
+1
Degrees
x=2.39358…∘+120∘n,x=57.60641…∘+120∘n,x=150∘+360∘n,x=210∘+360∘n
Solution steps
16sin(3x)cos(x)+83​sin(3x)−2cos(x)−3​=0
Factor 16sin(3x)cos(x)+83​sin(3x)−2cos(x)−3​:(−1+8sin(3x))(2cos(x)+3​)
16sin(3x)cos(x)+83​sin(3x)−2cos(x)−3​
Factor 16sin(3x)cos(x)−2cos(x):2cos(x)(8sin(3x)−1)
16sin(3x)cos(x)−2cos(x)
Rewrite as=8⋅2cos(x)sin(3x)−1⋅2cos(x)
Factor out common term 2cos(x)=2cos(x)(8sin(3x)−1)
Factor 83​sin(3x)−3​:3​(8sin(3x)−1)
83​sin(3x)−3​
Factor out common term 3​=3​(8sin(3x)−1)
=2cos(x)(8sin(3x)−1)+3​(8sin(3x)−1)
Factor out common term (−1+8sin(3x))=(−1+8sin(3x))(2cos(x)+3​)
(−1+8sin(3x))(2cos(x)+3​)=0
Solving each part separately−1+8sin(3x)=0or2cos(x)+3​=0
−1+8sin(3x)=0:x=3arcsin(81​)​+32πn​,x=3π​−3arcsin(81​)​+32πn​
−1+8sin(3x)=0
Move 1to the right side
−1+8sin(3x)=0
Add 1 to both sides−1+8sin(3x)+1=0+1
Simplify8sin(3x)=1
8sin(3x)=1
Divide both sides by 8
8sin(3x)=1
Divide both sides by 888sin(3x)​=81​
Simplifysin(3x)=81​
sin(3x)=81​
Apply trig inverse properties
sin(3x)=81​
General solutions for sin(3x)=81​sin(x)=a⇒x=arcsin(a)+2πn,x=π−arcsin(a)+2πn3x=arcsin(81​)+2πn,3x=π−arcsin(81​)+2πn
3x=arcsin(81​)+2πn,3x=π−arcsin(81​)+2πn
Solve 3x=arcsin(81​)+2πn:x=3arcsin(81​)​+32πn​
3x=arcsin(81​)+2πn
Divide both sides by 3
3x=arcsin(81​)+2πn
Divide both sides by 333x​=3arcsin(81​)​+32πn​
Simplifyx=3arcsin(81​)​+32πn​
x=3arcsin(81​)​+32πn​
Solve 3x=π−arcsin(81​)+2πn:x=3π​−3arcsin(81​)​+32πn​
3x=π−arcsin(81​)+2πn
Divide both sides by 3
3x=π−arcsin(81​)+2πn
Divide both sides by 333x​=3π​−3arcsin(81​)​+32πn​
Simplifyx=3π​−3arcsin(81​)​+32πn​
x=3π​−3arcsin(81​)​+32πn​
x=3arcsin(81​)​+32πn​,x=3π​−3arcsin(81​)​+32πn​
2cos(x)+3​=0:x=65π​+2πn,x=67π​+2πn
2cos(x)+3​=0
Move 3​to the right side
2cos(x)+3​=0
Subtract 3​ from both sides2cos(x)+3​−3​=0−3​
Simplify2cos(x)=−3​
2cos(x)=−3​
Divide both sides by 2
2cos(x)=−3​
Divide both sides by 222cos(x)​=2−3​​
Simplifycos(x)=−23​​
cos(x)=−23​​
General solutions for cos(x)=−23​​
cos(x) periodicity table with 2πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​cos(x)123​​22​​21​0−21​−22​​−23​​​xπ67π​45π​34π​23π​35π​47π​611π​​cos(x)−1−23​​−22​​−21​021​22​​23​​​​
x=65π​+2πn,x=67π​+2πn
x=65π​+2πn,x=67π​+2πn
Combine all the solutionsx=3arcsin(81​)​+32πn​,x=3π​−3arcsin(81​)​+32πn​,x=65π​+2πn,x=67π​+2πn
Show solutions in decimal formx=30.12532…​+32πn​,x=3π​−30.12532…​+32πn​,x=65π​+2πn,x=67π​+2πn

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