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Trigonometry Examples
Step 1
Step 1.1
To find the coordinate of the vertex, set the inside of the absolute value equal to . In this case, .
Step 1.2
Solve the equation to find the coordinate for the absolute value vertex.
Step 1.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 1.2.2
Simplify the right side.
Step 1.2.2.1
The exact value of is .
Step 1.2.3
Divide each term in by and simplify.
Step 1.2.3.1
Divide each term in by .
Step 1.2.3.2
Simplify the left side.
Step 1.2.3.2.1
Cancel the common factor of .
Step 1.2.3.2.1.1
Cancel the common factor.
Step 1.2.3.2.1.2
Divide by .
Step 1.2.3.3
Simplify the right side.
Step 1.2.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.3.3.2
Multiply .
Step 1.2.3.3.2.1
Multiply by .
Step 1.2.3.3.2.2
Multiply by .
Step 1.2.4
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 1.2.5
Solve for .
Step 1.2.5.1
Simplify.
Step 1.2.5.1.1
To write as a fraction with a common denominator, multiply by .
Step 1.2.5.1.2
Combine and .
Step 1.2.5.1.3
Combine the numerators over the common denominator.
Step 1.2.5.1.4
Multiply by .
Step 1.2.5.1.5
Subtract from .
Step 1.2.5.2
Divide each term in by and simplify.
Step 1.2.5.2.1
Divide each term in by .
Step 1.2.5.2.2
Simplify the left side.
Step 1.2.5.2.2.1
Cancel the common factor of .
Step 1.2.5.2.2.1.1
Cancel the common factor.
Step 1.2.5.2.2.1.2
Divide by .
Step 1.2.5.2.3
Simplify the right side.
Step 1.2.5.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.2.5.2.3.2
Multiply .
Step 1.2.5.2.3.2.1
Multiply by .
Step 1.2.5.2.3.2.2
Multiply by .
Step 1.2.6
Find the period of .
Step 1.2.6.1
The period of the function can be calculated using .
Step 1.2.6.2
Replace with in the formula for period.
Step 1.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.2.6.4
Cancel the common factor of .
Step 1.2.6.4.1
Cancel the common factor.
Step 1.2.6.4.2
Divide by .
Step 1.2.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 1.2.8
Consolidate the answers.
, for any integer
, for any integer
Step 1.3
Replace the variable with in the expression.
Step 1.4
Simplify each term.
Step 1.4.1
Apply the distributive property.
Step 1.4.2
Cancel the common factor of .
Step 1.4.2.1
Factor out of .
Step 1.4.2.2
Cancel the common factor.
Step 1.4.2.3
Rewrite the expression.
Step 1.4.3
Cancel the common factor of .
Step 1.4.3.1
Cancel the common factor.
Step 1.4.3.2
Rewrite the expression.
Step 1.5
The absolute value vertex is .
Step 2
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation:
Set-Builder Notation:
Step 3
The absolute value can be graphed using the points around the vertex
Step 4