Trigonometry Examples

Solve for x csc(x)^2=(-1/5)^2+1
Step 1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2
Simplify .
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Step 2.1
Use the power rule to distribute the exponent.
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Step 2.1.1
Apply the product rule to .
Step 2.1.2
Apply the product rule to .
Step 2.2
Raise to the power of .
Step 2.3
Multiply by .
Step 2.4
One to any power is one.
Step 2.5
Raise to the power of .
Step 2.6
Write as a fraction with a common denominator.
Step 2.7
Combine the numerators over the common denominator.
Step 2.8
Add and .
Step 2.9
Rewrite as .
Step 2.10
Simplify the denominator.
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Step 2.10.1
Rewrite as .
Step 2.10.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.1
First, use the positive value of the to find the first solution.
Step 3.2
Next, use the negative value of the to find the second solution.
Step 3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Set up each of the solutions to solve for .
Step 5
Solve for in .
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Step 5.1
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
Step 5.2
Simplify the right side.
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Step 5.2.1
Evaluate .
Step 5.3
The cosecant function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 5.4
Solve for .
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Step 5.4.1
Remove parentheses.
Step 5.4.2
Remove parentheses.
Step 5.4.3
Subtract from .
Step 5.5
Find the period of .
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Step 5.5.1
The period of the function can be calculated using .
Step 5.5.2
Replace with in the formula for period.
Step 5.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.5.4
Divide by .
Step 5.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 6
Solve for in .
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Step 6.1
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
Step 6.2
Simplify the right side.
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Step 6.2.1
Evaluate .
Step 6.3
The cosecant function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Step 6.4
Simplify the expression to find the second solution.
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Step 6.4.1
Subtract from .
Step 6.4.2
The resulting angle of is positive, less than , and coterminal with .
Step 6.5
Find the period of .
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Step 6.5.1
The period of the function can be calculated using .
Step 6.5.2
Replace with in the formula for period.
Step 6.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.5.4
Divide by .
Step 6.6
Add to every negative angle to get positive angles.
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Step 6.6.1
Add to to find the positive angle.
Step 6.6.2
Subtract from .
Step 6.6.3
List the new angles.
Step 6.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 7
List all of the solutions.
, for any integer
Step 8
Consolidate the solutions.
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Step 8.1
Consolidate and to .
, for any integer
Step 8.2
Consolidate and to .
, for any integer
, for any integer