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Trigonometry Examples
Step 1
Step 1.1
Expand using the FOIL Method.
Step 1.1.1
Apply the distributive property.
Step 1.1.2
Apply the distributive property.
Step 1.1.3
Apply the distributive property.
Step 1.2
Simplify each term.
Step 1.2.1
Multiply by .
Step 1.2.2
Multiply by .
Step 1.2.3
Multiply by .
Step 2
Step 2.1
Factor out the greatest common factor from each group.
Step 2.1.1
Group the first two terms and the last two terms.
Step 2.1.2
Factor out the greatest common factor (GCF) from each group.
Step 2.2
Factor the polynomial by factoring out the greatest common factor, .
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
Step 4.1
Set equal to .
Step 4.2
Solve for .
Step 4.2.1
Subtract from both sides of the equation.
Step 4.2.2
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
Step 4.2.3
Simplify the right side.
Step 4.2.3.1
The exact value of is .
Step 4.2.4
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 4.2.5
Simplify the expression to find the second solution.
Step 4.2.5.1
Subtract from .
Step 4.2.5.2
The resulting angle of is positive, less than , and coterminal with .
Step 4.2.6
Find the period of .
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
Add to every negative angle to get positive angles.
Step 4.2.7.1
Add to to find the positive angle.
Step 4.2.7.2
To write as a fraction with a common denominator, multiply by .
Step 4.2.7.3
Combine fractions.
Step 4.2.7.3.1
Combine and .
Step 4.2.7.3.2
Combine the numerators over the common denominator.
Step 4.2.7.4
Simplify the numerator.
Step 4.2.7.4.1
Multiply by .
Step 4.2.7.4.2
Subtract from .
Step 4.2.7.5
List the new angles.
Step 4.2.8
The period of the function is so values will repeat every radians in both directions.
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, for any integer
Step 5
Step 5.1
Set equal to .
Step 5.2
Solve for .
Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 5.2.3
Simplify the right side.
Step 5.2.3.1
The exact value of is .
Step 5.2.4
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 5.2.5
Simplify the expression to find the second solution.
Step 5.2.5.1
Add to .
Step 5.2.5.2
The resulting angle of is positive and coterminal with .
Step 5.2.6
Find the period of .
Step 5.2.6.1
The period of the function can be calculated using .
Step 5.2.6.2
Replace with in the formula for period.
Step 5.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.6.4
Divide by .
Step 5.2.7
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 6
The final solution is all the values that make true.
, for any integer
Step 7
Consolidate and to .
, for any integer