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Trigonometry Examples
Step 1
Replace the with based on the identity.
Step 2
Reorder the polynomial.
Step 3
Substitute for .
Step 4
Add to both sides of the equation.
Step 5
Add and .
Step 6
Step 6.1
Rewrite as .
Step 6.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 6.3
Rewrite the polynomial.
Step 6.4
Factor using the perfect square trinomial rule , where and .
Step 7
Set the equal to .
Step 8
Subtract from both sides of the equation.
Step 9
Substitute for .
Step 10
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
Step 11
Step 11.1
The exact value of is .
Step 12
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 13
Step 13.1
Subtract from .
Step 13.2
The resulting angle of is positive, less than , and coterminal with .
Step 14
Step 14.1
The period of the function can be calculated using .
Step 14.2
Replace with in the formula for period.
Step 14.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 14.4
Divide by .
Step 15
Step 15.1
Add to to find the positive angle.
Step 15.2
To write as a fraction with a common denominator, multiply by .
Step 15.3
Combine fractions.
Step 15.3.1
Combine and .
Step 15.3.2
Combine the numerators over the common denominator.
Step 15.4
Simplify the numerator.
Step 15.4.1
Multiply by .
Step 15.4.2
Subtract from .
Step 15.5
List the new angles.
Step 16
The period of the function is so values will repeat every radians in both directions.
, for any integer