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Trigonometry Examples
Step 1
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 2
Step 2.1
The exact value of is .
Step 3
Multiply both sides of the equation by .
Step 4
Step 4.1
Simplify the left side.
Step 4.1.1
Cancel the common factor of .
Step 4.1.1.1
Cancel the common factor.
Step 4.1.1.2
Rewrite the expression.
Step 4.2
Simplify the right side.
Step 4.2.1
Combine and .
Step 5
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 6
Step 6.1
Multiply both sides of the equation by .
Step 6.2
Simplify both sides of the equation.
Step 6.2.1
Simplify the left side.
Step 6.2.1.1
Cancel the common factor of .
Step 6.2.1.1.1
Cancel the common factor.
Step 6.2.1.1.2
Rewrite the expression.
Step 6.2.2
Simplify the right side.
Step 6.2.2.1
Simplify .
Step 6.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.1.2
Combine fractions.
Step 6.2.2.1.2.1
Combine and .
Step 6.2.2.1.2.2
Combine the numerators over the common denominator.
Step 6.2.2.1.3
Simplify the numerator.
Step 6.2.2.1.3.1
Move to the left of .
Step 6.2.2.1.3.2
Add and .
Step 6.2.2.1.4
Multiply .
Step 6.2.2.1.4.1
Combine and .
Step 6.2.2.1.4.2
Multiply by .
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
is approximately which is positive so remove the absolute value
Step 7.4
Multiply the numerator by the reciprocal of the denominator.
Step 7.5
Move to the left of .
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 9
Consolidate the answers.
, for any integer