Trigonometry Examples

Solve for x cot(x)^2-8cot(x)+12=0
Step 1
Factor the left side of the equation.
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Step 1.1
Let . Substitute for all occurrences of .
Step 1.2
Factor using the AC method.
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Step 1.2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 1.2.2
Write the factored form using these integers.
Step 1.3
Replace all occurrences of with .
Step 2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3
Set equal to and solve for .
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Step 3.1
Set equal to .
Step 3.2
Solve for .
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Step 3.2.1
Add to both sides of the equation.
Step 3.2.2
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 3.2.3
Simplify the right side.
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Step 3.2.3.1
Evaluate .
Step 3.2.4
The cotangent 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 3.2.5
Solve for .
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Step 3.2.5.1
Remove parentheses.
Step 3.2.5.2
Remove parentheses.
Step 3.2.5.3
Add and .
Step 3.2.6
Find the period of .
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Step 3.2.6.1
The period of the function can be calculated using .
Step 3.2.6.2
Replace with in the formula for period.
Step 3.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.6.4
Divide by .
Step 3.2.7
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 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
Add to both sides of the equation.
Step 4.2.2
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 4.2.3
Simplify the right side.
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Step 4.2.3.1
Evaluate .
Step 4.2.4
The cotangent 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 4.2.5
Solve for .
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Step 4.2.5.1
Remove parentheses.
Step 4.2.5.2
Remove parentheses.
Step 4.2.5.3
Add and .
Step 4.2.6
Find the period of .
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Step 4.2.6.1
The period of the function can be calculated using .
Step 4.2.6.2
Replace with in the formula for period.
Step 4.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.6.4
Divide by .
Step 4.2.7
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
The final solution is all the values that make true.
, for any integer
Step 6
Consolidate the answers.
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Step 6.1
Consolidate and to .
, for any integer
Step 6.2
Consolidate and to .
, for any integer
, for any integer