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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
Step 3.1
Add to both sides of the inequality.
Step 3.2
To write as a fraction with a common denominator, multiply by .
Step 3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 3.3.1
Multiply by .
Step 3.3.2
Multiply by .
Step 3.4
Combine the numerators over the common denominator.
Step 3.5
Simplify the numerator.
Step 3.5.1
Move to the left of .
Step 3.5.2
Add and .
Step 4
Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
Step 4.2.1
Cancel the common factor of .
Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
Step 4.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.2
Cancel the common factor of .
Step 4.3.2.1
Factor out of .
Step 4.3.2.2
Cancel the common factor.
Step 4.3.2.3
Rewrite the expression.
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
Simplify .
Step 6.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.1.2
Combine fractions.
Step 6.1.2.1
Combine and .
Step 6.1.2.2
Combine the numerators over the common denominator.
Step 6.1.3
Simplify the numerator.
Step 6.1.3.1
Move to the left of .
Step 6.1.3.2
Add and .
Step 6.2
Move all terms not containing to the right side of the equation.
Step 6.2.1
Add to both sides of the equation.
Step 6.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.2.3.1
Multiply by .
Step 6.2.3.2
Multiply by .
Step 6.2.4
Combine the numerators over the common denominator.
Step 6.2.5
Simplify the numerator.
Step 6.2.5.1
Move to the left of .
Step 6.2.5.2
Add and .
Step 6.3
Divide each term in by and simplify.
Step 6.3.1
Divide each term in by .
Step 6.3.2
Simplify the left side.
Step 6.3.2.1
Cancel the common factor of .
Step 6.3.2.1.1
Cancel the common factor.
Step 6.3.2.1.2
Divide by .
Step 6.3.3
Simplify the right side.
Step 6.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.3.3.2
Multiply .
Step 6.3.3.2.1
Multiply by .
Step 6.3.3.2.2
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
The absolute value is the distance between a number and zero. The distance between and is .
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
Step 10
Step 10.1
Set the argument in equal to to find where the expression is undefined.
, for any integer
Step 10.2
Solve for .
Step 10.2.1
Move all terms not containing to the right side of the equation.
Step 10.2.1.1
Add to both sides of the equation.
Step 10.2.1.2
Combine the numerators over the common denominator.
Step 10.2.1.3
Add and .
Step 10.2.1.4
Cancel the common factor of .
Step 10.2.1.4.1
Cancel the common factor.
Step 10.2.1.4.2
Divide by .
Step 10.2.2
Divide each term in by and simplify.
Step 10.2.2.1
Divide each term in by .
Step 10.2.2.2
Simplify the left side.
Step 10.2.2.2.1
Cancel the common factor of .
Step 10.2.2.2.1.1
Cancel the common factor.
Step 10.2.2.2.1.2
Divide by .
Step 10.3
The domain is all values of that make the expression defined.
, for any integer
, 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
Test a value on the interval to see if it makes the inequality true.
Step 12.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.2.2
Replace with in the original inequality.
Step 12.2.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 12.3
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
False
Step 13
The solution consists of all of the true intervals.
, for any integer
Step 14
Convert the inequality to interval notation.
Step 15