Trigonometry Examples

Solve for x 9sin(x)^2tan(x)=16tan(x)
Step 1
Subtract from both sides of the equation.
Step 2
Factor .
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Step 2.1
Factor out of .
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Step 2.1.1
Factor out of .
Step 2.1.2
Factor out of .
Step 2.1.3
Factor out of .
Step 2.2
Rewrite as .
Step 2.3
Rewrite as .
Step 2.4
Factor.
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Step 2.4.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.4.2
Remove unnecessary parentheses.
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Solve for .
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Step 4.2.1
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 4.2.2
Simplify the right side.
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Step 4.2.2.1
The exact value of is .
Step 4.2.3
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 4.2.4
Add and .
Step 4.2.5
Find the period of .
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Step 4.2.5.1
The period of the function can be calculated using .
Step 4.2.5.2
Replace with in the formula for period.
Step 4.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.5.4
Divide by .
Step 4.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 5
Set equal to and solve for .
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Step 5.1
Set equal to .
Step 5.2
Solve for .
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Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Divide each term in by and simplify.
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Step 5.2.2.1
Divide each term in by .
Step 5.2.2.2
Simplify the left side.
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Step 5.2.2.2.1
Cancel the common factor of .
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Step 5.2.2.2.1.1
Cancel the common factor.
Step 5.2.2.2.1.2
Divide by .
Step 5.2.2.3
Simplify the right side.
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Step 5.2.2.3.1
Move the negative in front of the fraction.
Step 5.2.3
The range of sine is . Since does not fall in this range, there is no solution.
No solution
No solution
No solution
Step 6
The final solution is all the values that make true.
, for any integer
Step 7
Consolidate the answers.
, for any integer