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Trigonometry Examples
Step 1
Divide each term in the equation by .
Step 2
Convert from to .
Step 3
Step 3.1
Cancel the common factor.
Step 3.2
Rewrite the expression.
Step 4
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 5
Step 5.1
The exact value of is .
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
To write as a fraction with a common denominator, multiply by .
Step 7.2
Combine fractions.
Step 7.2.1
Combine and .
Step 7.2.2
Combine the numerators over the common denominator.
Step 7.3
Simplify the numerator.
Step 7.3.1
Move to the left of .
Step 7.3.2
Add and .
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 8.4
Divide by .
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 10
Consolidate the answers.
, for any integer
Step 11
Use each root to create test intervals.
Step 12
Step 12.1
Test a value on the interval to see if it makes the inequality true.
Step 12.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.1.2
Replace with in the original inequality.
Step 12.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 12.2
Compare the intervals to determine which ones satisfy the original inequality.
True
True
Step 13
The solution consists of all of the true intervals.
, for any integer
Step 14