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Trigonometry Examples
Step 1
Step 1.1
Rewrite the equation as .
Step 1.2
Multiply both sides of the equation by .
Step 1.3
Simplify both sides of the equation.
Step 1.3.1
Simplify the left side.
Step 1.3.1.1
Simplify .
Step 1.3.1.1.1
Apply the distributive property.
Step 1.3.1.1.2
Combine and .
Step 1.3.1.1.3
Cancel the common factor of .
Step 1.3.1.1.3.1
Cancel the common factor.
Step 1.3.1.1.3.2
Rewrite the expression.
Step 1.3.1.1.4
Apply the distributive property.
Step 1.3.1.1.5
Cancel the common factor of .
Step 1.3.1.1.5.1
Cancel the common factor.
Step 1.3.1.1.5.2
Rewrite the expression.
Step 1.3.1.1.6
Cancel the common factor of .
Step 1.3.1.1.6.1
Factor out of .
Step 1.3.1.1.6.2
Cancel the common factor.
Step 1.3.1.1.6.3
Rewrite the expression.
Step 1.3.1.1.7
Cancel the common factor of .
Step 1.3.1.1.7.1
Cancel the common factor.
Step 1.3.1.1.7.2
Rewrite the expression.
Step 1.3.2
Simplify the right side.
Step 1.3.2.1
Combine and .
Step 1.4
Subtract from both sides of the equation.
Step 1.5
Simplify .
Step 1.5.1
To write as a fraction with a common denominator, multiply by .
Step 1.5.2
Simplify terms.
Step 1.5.2.1
Combine and .
Step 1.5.2.2
Combine the numerators over the common denominator.
Step 1.5.3
Simplify the numerator.
Step 1.5.3.1
Factor out of .
Step 1.5.3.1.1
Factor out of .
Step 1.5.3.1.2
Factor out of .
Step 1.5.3.1.3
Factor out of .
Step 1.5.3.2
Rewrite as .
Step 1.5.3.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.5.3.4
Simplify.
Step 1.5.3.4.1
Add and .
Step 1.5.3.4.2
Subtract from .
Step 2
Step 2.1
Rewrite the equation in vertex form.
Step 2.1.1
Complete the square for .
Step 2.1.1.1
Use the form , to find the values of , , and .
Step 2.1.1.2
Consider the vertex form of a parabola.
Step 2.1.1.3
Find the value of using the formula .
Step 2.1.1.3.1
Substitute the values of and into the formula .
Step 2.1.1.3.2
Cancel the common factor of and .
Step 2.1.1.3.2.1
Factor out of .
Step 2.1.1.3.2.2
Cancel the common factors.
Step 2.1.1.3.2.2.1
Factor out of .
Step 2.1.1.3.2.2.2
Cancel the common factor.
Step 2.1.1.3.2.2.3
Rewrite the expression.
Step 2.1.1.3.2.2.4
Divide by .
Step 2.1.1.4
Find the value of using the formula .
Step 2.1.1.4.1
Substitute the values of , and into the formula .
Step 2.1.1.4.2
Simplify the right side.
Step 2.1.1.4.2.1
Simplify each term.
Step 2.1.1.4.2.1.1
Raising to any positive power yields .
Step 2.1.1.4.2.1.2
Multiply by .
Step 2.1.1.4.2.1.3
Divide by .
Step 2.1.1.4.2.1.4
Multiply by .
Step 2.1.1.4.2.2
Add and .
Step 2.1.1.5
Substitute the values of , , and into the vertex form .
Step 2.1.2
Set equal to the new right side.
Step 2.2
Use the vertex form, , to determine the values of , , and .
Step 2.3
Since the value of is positive, the parabola opens up.
Opens Up
Step 2.4
Find the vertex .
Step 2.5
Find , the distance from the vertex to the focus.
Step 2.5.1
Find the distance from the vertex to a focus of the parabola by using the following formula.
Step 2.5.2
Substitute the value of into the formula.
Step 2.5.3
Cancel the common factor of .
Step 2.5.3.1
Cancel the common factor.
Step 2.5.3.2
Rewrite the expression.
Step 2.6
Find the focus.
Step 2.6.1
The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Step 2.6.2
Substitute the known values of , , and into the formula and simplify.
Step 2.7
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Step 2.8
Find the directrix.
Step 2.8.1
The directrix of a parabola is the horizontal line found by subtracting from the y-coordinate of the vertex if the parabola opens up or down.
Step 2.8.2
Substitute the known values of and into the formula and simplify.
Step 2.9
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 3
Step 3.1
Replace the variable with in the expression.
Step 3.2
Simplify the result.
Step 3.2.1
Simplify the numerator.
Step 3.2.1.1
Add and .
Step 3.2.1.2
Combine exponents.
Step 3.2.1.2.1
Multiply by .
Step 3.2.1.2.2
Multiply by .
Step 3.2.2
Divide by .
Step 3.2.3
The final answer is .
Step 3.3
The value at is .
Step 3.4
Replace the variable with in the expression.
Step 3.5
Simplify the result.
Step 3.5.1
Simplify the numerator.
Step 3.5.1.1
Add and .
Step 3.5.1.2
Multiply by .
Step 3.5.1.3
Subtract from .
Step 3.5.2
Simplify the expression.
Step 3.5.2.1
Multiply by .
Step 3.5.2.2
Move the negative in front of the fraction.
Step 3.5.3
The final answer is .
Step 3.6
The value at is .
Step 3.7
Replace the variable with in the expression.
Step 3.8
Simplify the result.
Step 3.8.1
Simplify the numerator.
Step 3.8.1.1
Add and .
Step 3.8.1.2
Multiply by .
Step 3.8.1.3
Subtract from .
Step 3.8.2
Simplify the expression.
Step 3.8.2.1
Multiply by .
Step 3.8.2.2
Divide by .
Step 3.8.3
The final answer is .
Step 3.9
The value at is .
Step 3.10
Replace the variable with in the expression.
Step 3.11
Simplify the result.
Step 3.11.1
Simplify the numerator.
Step 3.11.1.1
Add and .
Step 3.11.1.2
Multiply by .
Step 3.11.1.3
Subtract from .
Step 3.11.2
Simplify the expression.
Step 3.11.2.1
Multiply by .
Step 3.11.2.2
Move the negative in front of the fraction.
Step 3.11.3
The final answer is .
Step 3.12
The value at is .
Step 3.13
Graph the parabola using its properties and the selected points.
Step 4
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Step 5