Trigonometry Examples

Solve for x cot(x)^2+6cot(x)-2=0
Step 1
Substitute for .
Step 2
Use the quadratic formula to find the solutions.
Step 3
Substitute the values , , and into the quadratic formula and solve for .
Step 4
Simplify.
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Step 4.1
Simplify the numerator.
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Step 4.1.1
Raise to the power of .
Step 4.1.2
Multiply .
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Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply by .
Step 4.1.3
Add and .
Step 4.1.4
Rewrite as .
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Step 4.1.4.1
Factor out of .
Step 4.1.4.2
Rewrite as .
Step 4.1.5
Pull terms out from under the radical.
Step 4.2
Multiply by .
Step 4.3
Simplify .
Step 5
The final answer is the combination of both solutions.
Step 6
Substitute for .
Step 7
Set up each of the solutions to solve for .
Step 8
Solve for in .
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Step 8.1
Convert the right side of the equation to its decimal equivalent.
Step 8.2
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 8.3
Simplify the right side.
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Step 8.3.1
Evaluate .
Step 8.4
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 8.5
Solve for .
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Step 8.5.1
Remove parentheses.
Step 8.5.2
Remove parentheses.
Step 8.5.3
Add and .
Step 8.6
Find the period of .
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Step 8.6.1
The period of the function can be calculated using .
Step 8.6.2
Replace with in the formula for period.
Step 8.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 8.6.4
Divide by .
Step 8.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 9
Solve for in .
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Step 9.1
Convert the right side of the equation to its decimal equivalent.
Step 9.2
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 9.3
Simplify the right side.
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Step 9.3.1
Evaluate .
Step 9.4
The cotangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 9.5
Simplify the expression to find the second solution.
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Step 9.5.1
Add to .
Step 9.5.2
The resulting angle of is positive and coterminal with .
Step 9.6
Find the period of .
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Step 9.6.1
The period of the function can be calculated using .
Step 9.6.2
Replace with in the formula for period.
Step 9.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 9.6.4
Divide by .
Step 9.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 10
List all of the solutions.
, for any integer
Step 11
Consolidate the solutions.
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Step 11.1
Consolidate and to .
, for any integer
Step 11.2
Consolidate and to .
, for any integer
, for any integer