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Trigonometry Examples
Step 1
Replace the with based on the identity.
Step 2
Step 2.1
Apply the distributive property.
Step 2.2
Multiply by .
Step 3
Reorder the polynomial.
Step 4
Substitute for .
Step 5
Step 5.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 5.1.1
Factor out of .
Step 5.1.2
Rewrite as plus
Step 5.1.3
Apply the distributive property.
Step 5.1.4
Multiply by .
Step 5.2
Factor out the greatest common factor from each group.
Step 5.2.1
Group the first two terms and the last two terms.
Step 5.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.3
Factor the polynomial by factoring out the greatest common factor, .
Step 6
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 7
Step 7.1
Set equal to .
Step 7.2
Solve for .
Step 7.2.1
Subtract from both sides of the equation.
Step 7.2.2
Divide each term in by and simplify.
Step 7.2.2.1
Divide each term in by .
Step 7.2.2.2
Simplify the left side.
Step 7.2.2.2.1
Cancel the common factor of .
Step 7.2.2.2.1.1
Cancel the common factor.
Step 7.2.2.2.1.2
Divide by .
Step 7.2.2.3
Simplify the right side.
Step 7.2.2.3.1
Move the negative in front of the fraction.
Step 8
Step 8.1
Set equal to .
Step 8.2
Add to both sides of the equation.
Step 9
The final solution is all the values that make true.
Step 10
Substitute for .
Step 11
Set up each of the solutions to solve for .
Step 12
Step 12.1
The range of cosecant is and . Since does not fall in this range, there is no solution.
No solution
No solution
Step 13
Step 13.1
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
Step 13.2
Simplify the right side.
Step 13.2.1
The exact value of is .
Step 13.3
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 13.4
Simplify .
Step 13.4.1
To write as a fraction with a common denominator, multiply by .
Step 13.4.2
Combine fractions.
Step 13.4.2.1
Combine and .
Step 13.4.2.2
Combine the numerators over the common denominator.
Step 13.4.3
Simplify the numerator.
Step 13.4.3.1
Move to the left of .
Step 13.4.3.2
Subtract from .
Step 13.5
Find the period of .
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
List all of the solutions.
, for any integer