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Popular Trigonometry >

((sin(x)+tan(x)))/(1+cos(x))=m*sec(x)

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Solution

1+cos(x)(sin(x)+tan(x))​=m⋅sec(x)

Solution

x=arcsin(m)+2πn,x=π+arcsin(−m)+2πn
Solution steps
1+cos(x)(sin(x)+tan(x))​=msec(x)
Subtract msec(x) from both sides1+cos(x)sin(x)+tan(x)​−msec(x)=0
Simplify 1+cos(x)sin(x)+tan(x)​−msec(x):1+cos(x)sin(x)+tan(x)−msec(x)(1+cos(x))​
1+cos(x)sin(x)+tan(x)​−msec(x)
Convert element to fraction: msec(x)=1+cos(x)msec(x)(1+cos(x))​=1+cos(x)sin(x)+tan(x)​−1+cos(x)msec(x)(1+cos(x))​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=1+cos(x)sin(x)+tan(x)−msec(x)(1+cos(x))​
1+cos(x)sin(x)+tan(x)−msec(x)(1+cos(x))​=0
g(x)f(x)​=0⇒f(x)=0sin(x)+tan(x)−msec(x)(1+cos(x))=0
Express with sin, cos
sin(x)+tan(x)−(1+cos(x))sec(x)m
Use the basic trigonometric identity: tan(x)=cos(x)sin(x)​=sin(x)+cos(x)sin(x)​−(1+cos(x))sec(x)m
Use the basic trigonometric identity: sec(x)=cos(x)1​=sin(x)+cos(x)sin(x)​−(1+cos(x))cos(x)1​m
Simplify sin(x)+cos(x)sin(x)​−(1+cos(x))cos(x)1​m:cos(x)sin(x)cos(x)+sin(x)−m(1+cos(x))​
sin(x)+cos(x)sin(x)​−(1+cos(x))cos(x)1​m
(1+cos(x))cos(x)1​m=cos(x)m(1+cos(x))​
(1+cos(x))cos(x)1​m
Multiply fractions: a⋅cb​=ca⋅b​=cos(x)1⋅(1+cos(x))m​
Multiply: 1⋅(1+cos(x))=(1+cos(x))=cos(x)m(cos(x)+1)​
=sin(x)+cos(x)sin(x)​−cos(x)m(cos(x)+1)​
Combine the fractions cos(x)sin(x)​−cos(x)m(cos(x)+1)​:cos(x)sin(x)−m(1+cos(x))​
Apply rule ca​±cb​=ca±b​=cos(x)sin(x)−m(cos(x)+1)​
=sin(x)+cos(x)sin(x)−m(cos(x)+1)​
Convert element to fraction: sin(x)=cos(x)sin(x)cos(x)​=cos(x)sin(x)cos(x)​+cos(x)sin(x)−(1+cos(x))m​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=cos(x)sin(x)cos(x)+sin(x)−(1+cos(x))m​
=cos(x)sin(x)cos(x)+sin(x)−m(1+cos(x))​
cos(x)sin(x)−(1+cos(x))m+cos(x)sin(x)​=0
g(x)f(x)​=0⇒f(x)=0sin(x)−(1+cos(x))m+cos(x)sin(x)=0
Factor sin(x)−(1+cos(x))m+cos(x)sin(x):(1+cos(x))(sin(x)−m)
sin(x)−(1+cos(x))m+cos(x)sin(x)
Factor out common term sin(x)=sin(x)(1+cos(x))−(1+cos(x))m
Factor out common term (1+cos(x))=(1+cos(x))(sin(x)−m)
(1+cos(x))(sin(x)−m)=0
Solving each part separately1+cos(x)=0orsin(x)−m=0
1+cos(x)=0:x=π+2πn
1+cos(x)=0
Move 1to the right side
1+cos(x)=0
Subtract 1 from both sides1+cos(x)−1=0−1
Simplifycos(x)=−1
cos(x)=−1
General solutions for cos(x)=−1
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=π+2πn
x=π+2πn
sin(x)−m=0:x=arcsin(m)+2πn,x=π+arcsin(−m)+2πn
sin(x)−m=0
Move mto the right side
sin(x)−m=0
Add m to both sidessin(x)−m+m=0+m
Simplifysin(x)=m
sin(x)=m
Apply trig inverse properties
sin(x)=m
General solutions for sin(x)=msin(x)=a⇒x=arcsin(a)+2πn,x=π+arcsin(a)+2πnx=arcsin(m)+2πn,x=π+arcsin(−m)+2πn
x=arcsin(m)+2πn,x=π+arcsin(−m)+2πn
Combine all the solutionsx=π+2πn,x=arcsin(m)+2πn,x=π+arcsin(−m)+2πn
Since the equation is undefined for:π+2πnx=arcsin(m)+2πn,x=π+arcsin(−m)+2πn

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