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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 arctangent of both sides of the equation to extract from inside the arctangent.
Step 1.2.2
Simplify the right side.
Step 1.2.2.1
The exact value of is .
Step 1.3
Replace the variable with in the expression.
Step 1.4
Simplify .
Step 1.4.1
The exact value of is .
Step 1.4.2
The absolute value is the distance between a number and zero. The distance between and is .
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
Step 3.1
Substitute the value into . In this case, the point is .
Step 3.1.1
Replace the variable with in the expression.
Step 3.1.2
Simplify the result.
Step 3.1.2.1
Evaluate .
Step 3.1.2.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.1.2.3
The final answer is .
Step 3.2
Substitute the value into . In this case, the point is .
Step 3.2.1
Replace the variable with in the expression.
Step 3.2.2
Simplify the result.
Step 3.2.2.1
The exact value of is .
Step 3.2.2.2
is approximately which is negative so negate and remove the absolute value
Step 3.2.2.3
The final answer is .
Step 3.3
Substitute the value into . In this case, the point is .
Step 3.3.1
Replace the variable with in the expression.
Step 3.3.2
Simplify the result.
Step 3.3.2.1
Evaluate .
Step 3.3.2.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.3.2.3
The final answer is .
Step 3.4
The absolute value can be graphed using the points around the vertex
Step 4