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Popular Trigonometry >

9sin^2(x)tan(x)=16tan(x)

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Solution

9sin2(x)tan(x)=16tan(x)

Solution

x=πn
+1
Degrees
x=0∘+180∘n
Solution steps
9sin2(x)tan(x)=16tan(x)
Subtract 16tan(x) from both sides9sin2(x)tan(x)−16tan(x)=0
Factor 9sin2(x)tan(x)−16tan(x):tan(x)(3sin(x)+4)(3sin(x)−4)
9sin2(x)tan(x)−16tan(x)
Factor out common term tan(x)=tan(x)(9sin2(x)−16)
Factor 9sin2(x)−16:(3sin(x)+4)(3sin(x)−4)
9sin2(x)−16
Rewrite 9sin2(x)−16 as (3sin(x))2−42
9sin2(x)−16
Rewrite 9 as 32=32sin2(x)−16
Rewrite 16 as 42=32sin2(x)−42
Apply exponent rule: ambm=(ab)m32sin2(x)=(3sin(x))2=(3sin(x))2−42
=(3sin(x))2−42
Apply Difference of Two Squares Formula: x2−y2=(x+y)(x−y)(3sin(x))2−42=(3sin(x)+4)(3sin(x)−4)=(3sin(x)+4)(3sin(x)−4)
=tan(x)(3sin(x)+4)(3sin(x)−4)
tan(x)(3sin(x)+4)(3sin(x)−4)=0
Solving each part separatelytan(x)=0or3sin(x)+4=0or3sin(x)−4=0
tan(x)=0:x=πn
tan(x)=0
General solutions for tan(x)=0
tan(x) periodicity table with πn cycle:
x06π​4π​3π​2π​32π​43π​65π​​tan(x)033​​13​±∞−3​−1−33​​​​
x=0+πn
x=0+πn
Solve x=0+πn:x=πn
x=0+πn
0+πn=πnx=πn
x=πn
3sin(x)+4=0:No Solution
3sin(x)+4=0
Move 4to the right side
3sin(x)+4=0
Subtract 4 from both sides3sin(x)+4−4=0−4
Simplify3sin(x)=−4
3sin(x)=−4
Divide both sides by 3
3sin(x)=−4
Divide both sides by 333sin(x)​=3−4​
Simplifysin(x)=−34​
sin(x)=−34​
−1≤sin(x)≤1NoSolution
3sin(x)−4=0:No Solution
3sin(x)−4=0
Move 4to the right side
3sin(x)−4=0
Add 4 to both sides3sin(x)−4+4=0+4
Simplify3sin(x)=4
3sin(x)=4
Divide both sides by 3
3sin(x)=4
Divide both sides by 333sin(x)​=34​
Simplifysin(x)=34​
sin(x)=34​
−1≤sin(x)≤1NoSolution
Combine all the solutionsx=πn

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Popular Examples

3sin(2x)=5cos^2(2x)-1sin(x)=2,sin^2(x)=0cos(θ)=-sqrt(2)-8cos(2x)=0,(-pi,pi)1.66=sin(x)

Frequently Asked Questions (FAQ)

  • What is the general solution for 9sin^2(x)tan(x)=16tan(x) ?

    The general solution for 9sin^2(x)tan(x)=16tan(x) is x=pin
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