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Trigonometry Examples
Step 1
Step 1.1
Simplify each term.
Step 1.1.1
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 1.1.1.1
Reorder and .
Step 1.1.1.2
Rewrite in terms of sines and cosines.
Step 1.1.1.3
Cancel the common factors.
Step 1.1.2
Convert from to .
Step 2
Subtract from both sides of the equation.
Step 3
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
Step 4
Step 4.1
The exact value of is .
Step 5
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
Step 6.1
Subtract from .
Step 6.2
The resulting angle of is positive, less than , and coterminal with .
Step 7
Step 7.1
The period of the function can be calculated using .
Step 7.2
Replace with in the formula for period.
Step 7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.4
Divide by .
Step 8
Step 8.1
Add to to find the positive angle.
Step 8.2
To write as a fraction with a common denominator, multiply by .
Step 8.3
Combine fractions.
Step 8.3.1
Combine and .
Step 8.3.2
Combine the numerators over the common denominator.
Step 8.4
Simplify the numerator.
Step 8.4.1
Multiply by .
Step 8.4.2
Subtract from .
Step 8.5
List the new angles.
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer