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Trigonometry Examples
Step 1
Divide each term in the equation by .
Step 2
Separate fractions.
Step 3
Convert from to .
Step 4
Divide by .
Step 5
Step 5.1
Cancel the common factor.
Step 5.2
Divide by .
Step 6
Step 6.1
Divide each term in by .
Step 6.2
Simplify the left side.
Step 6.2.1
Cancel the common factor of .
Step 6.2.1.1
Cancel the common factor.
Step 6.2.1.2
Divide by .
Step 6.3
Simplify the right side.
Step 6.3.1
Divide by .
Step 7
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 8
Step 8.1
The exact value of is .
Step 9
Multiply both sides of the equation by .
Step 10
Step 10.1
Simplify the left side.
Step 10.1.1
Cancel the common factor of .
Step 10.1.1.1
Cancel the common factor.
Step 10.1.1.2
Rewrite the expression.
Step 10.2
Simplify the right side.
Step 10.2.1
Simplify .
Step 10.2.1.1
Cancel the common factor of .
Step 10.2.1.1.1
Move the leading negative in into the numerator.
Step 10.2.1.1.2
Factor out of .
Step 10.2.1.1.3
Cancel the common factor.
Step 10.2.1.1.4
Rewrite the expression.
Step 10.2.1.2
Move the negative in front of the fraction.
Step 11
The tangent 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 12
Step 12.1
Add to .
Step 12.2
The resulting angle of is positive and coterminal with .
Step 12.3
Solve for .
Step 12.3.1
Multiply both sides of the equation by .
Step 12.3.2
Simplify both sides of the equation.
Step 12.3.2.1
Simplify the left side.
Step 12.3.2.1.1
Cancel the common factor of .
Step 12.3.2.1.1.1
Cancel the common factor.
Step 12.3.2.1.1.2
Rewrite the expression.
Step 12.3.2.2
Simplify the right side.
Step 12.3.2.2.1
Cancel the common factor of .
Step 12.3.2.2.1.1
Factor out of .
Step 12.3.2.2.1.2
Cancel the common factor.
Step 12.3.2.2.1.3
Rewrite the expression.
Step 13
Step 13.1
The period of the function can be calculated using .
Step 13.2
Replace with in the formula for period.
Step 13.3
is approximately which is positive so remove the absolute value
Step 13.4
Multiply the numerator by the reciprocal of the denominator.
Step 13.5
Move to the left of .
Step 14
Step 14.1
Add to to find the positive angle.
Step 14.2
To write as a fraction with a common denominator, multiply by .
Step 14.3
Combine fractions.
Step 14.3.1
Combine and .
Step 14.3.2
Combine the numerators over the common denominator.
Step 14.4
Simplify the numerator.
Step 14.4.1
Multiply by .
Step 14.4.2
Subtract from .
Step 14.5
List the new angles.
Step 15
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 16
Consolidate the answers.
, for any integer