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Trigonometry Examples
Step 1
Step 1.1
Simplify the numerator.
Step 1.1.1
Write as a fraction with a common denominator.
Step 1.1.2
Combine the numerators over the common denominator.
Step 1.1.3
Subtract from .
Step 1.1.4
Rewrite as .
Step 1.1.5
Multiply by .
Step 1.1.6
Combine and simplify the denominator.
Step 1.1.6.1
Multiply by .
Step 1.1.6.2
Raise to the power of .
Step 1.1.6.3
Raise to the power of .
Step 1.1.6.4
Use the power rule to combine exponents.
Step 1.1.6.5
Add and .
Step 1.1.6.6
Rewrite as .
Step 1.1.6.6.1
Use to rewrite as .
Step 1.1.6.6.2
Apply the power rule and multiply exponents, .
Step 1.1.6.6.3
Combine and .
Step 1.1.6.6.4
Cancel the common factor of .
Step 1.1.6.6.4.1
Cancel the common factor.
Step 1.1.6.6.4.2
Rewrite the expression.
Step 1.1.6.6.5
Evaluate the exponent.
Step 1.1.7
Simplify the numerator.
Step 1.1.7.1
Combine using the product rule for radicals.
Step 1.1.7.2
Multiply by .
Step 1.2
Simplify the denominator.
Step 1.2.1
Write as a fraction with a common denominator.
Step 1.2.2
Combine the numerators over the common denominator.
Step 1.2.3
Subtract from .
Step 1.3
Multiply the numerator by the reciprocal of the denominator.
Step 1.4
Cancel the common factor of .
Step 1.4.1
Cancel the common factor.
Step 1.4.2
Rewrite the expression.
Step 1.5
Combine and .
Step 2
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 3
Step 3.1
Evaluate .
Step 4
Multiply both sides of the equation by .
Step 5
Step 5.1
Simplify the left side.
Step 5.1.1
Cancel the common factor of .
Step 5.1.1.1
Cancel the common factor.
Step 5.1.1.2
Rewrite the expression.
Step 5.2
Simplify the right side.
Step 5.2.1
Multiply by .
Step 6
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 7
Step 7.1
Multiply both sides of the equation by .
Step 7.2
Simplify both sides of the equation.
Step 7.2.1
Simplify the left side.
Step 7.2.1.1
Cancel the common factor of .
Step 7.2.1.1.1
Cancel the common factor.
Step 7.2.1.1.2
Rewrite the expression.
Step 7.2.2
Simplify the right side.
Step 7.2.2.1
Simplify .
Step 7.2.2.1.1
Add and .
Step 7.2.2.1.2
Multiply by .
Step 8
Step 8.1
The period of the function can be calculated using .
Step 8.2
Replace with in the formula for period.
Step 8.3
is approximately which is positive so remove the absolute value
Step 8.4
Multiply the numerator by the reciprocal of the denominator.
Step 8.5
Move to the left of .
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 10
Consolidate and to .
, for any integer