Trigonometry Examples

Solve for ? csc(x)^2=8cot(x)+10
Step 1
Replace the with based on the identity.
Step 2
Substitute for .
Step 3
Subtract from both sides of the equation.
Step 4
Subtract from both sides of the equation.
Step 5
Subtract from .
Step 6
Factor using the AC method.
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Step 6.1
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 6.2
Write the factored form using these integers.
Step 7
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 8
Set equal to and solve for .
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Step 8.1
Set equal to .
Step 8.2
Add to both sides of the equation.
Step 9
Set equal to and solve for .
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Step 9.1
Set equal to .
Step 9.2
Subtract from both sides of the equation.
Step 10
The final solution is all the values that make true.
Step 11
Substitute for .
Step 12
Set up each of the solutions to solve for .
Step 13
Solve for in .
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Step 13.1
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 13.2
Simplify the right side.
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Step 13.2.1
Evaluate .
Step 13.3
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 13.4
Solve for .
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Step 13.4.1
Remove parentheses.
Step 13.4.2
Remove parentheses.
Step 13.4.3
Add and .
Step 13.5
Find the period of .
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Step 13.5.1
The period of the function can be calculated using .
Step 13.5.2
Replace with in the formula for period.
Step 13.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 13.5.4
Divide by .
Step 13.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 14
Solve for in .
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Step 14.1
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 14.2
Simplify the right side.
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Step 14.2.1
The exact value of is .
Step 14.3
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 14.4
Simplify the expression to find the second solution.
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Step 14.4.1
Add to .
Step 14.4.2
The resulting angle of is positive and coterminal with .
Step 14.5
Find the period of .
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Step 14.5.1
The period of the function can be calculated using .
Step 14.5.2
Replace with in the formula for period.
Step 14.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 14.5.4
Divide by .
Step 14.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 15
List all of the solutions.
, for any integer
Step 16
Consolidate the solutions.
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Step 16.1
Consolidate and to .
, for any integer
Step 16.2
Consolidate and to .
, for any integer
, for any integer