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Trigonometry Examples
Step 1
Step 1.1
Subtract from both sides of the equation.
Step 1.2
Complete the square for .
Step 1.2.1
Use the form , to find the values of , , and .
Step 1.2.2
Consider the vertex form of a parabola.
Step 1.2.3
Find the value of using the formula .
Step 1.2.3.1
Substitute the values of and into the formula .
Step 1.2.3.2
Simplify the right side.
Step 1.2.3.2.1
Cancel the common factor of and .
Step 1.2.3.2.1.1
Factor out of .
Step 1.2.3.2.1.2
Cancel the common factors.
Step 1.2.3.2.1.2.1
Factor out of .
Step 1.2.3.2.1.2.2
Cancel the common factor.
Step 1.2.3.2.1.2.3
Rewrite the expression.
Step 1.2.3.2.2
Cancel the common factor of and .
Step 1.2.3.2.2.1
Factor out of .
Step 1.2.3.2.2.2
Cancel the common factors.
Step 1.2.3.2.2.2.1
Factor out of .
Step 1.2.3.2.2.2.2
Cancel the common factor.
Step 1.2.3.2.2.2.3
Rewrite the expression.
Step 1.2.3.2.2.2.4
Divide by .
Step 1.2.4
Find the value of using the formula .
Step 1.2.4.1
Substitute the values of , and into the formula .
Step 1.2.4.2
Simplify the right side.
Step 1.2.4.2.1
Simplify each term.
Step 1.2.4.2.1.1
Raise to the power of .
Step 1.2.4.2.1.2
Multiply by .
Step 1.2.4.2.1.3
Divide by .
Step 1.2.4.2.1.4
Multiply by .
Step 1.2.4.2.2
Subtract from .
Step 1.2.5
Substitute the values of , , and into the vertex form .
Step 1.3
Substitute for in the equation .
Step 1.4
Move to the right side of the equation by adding to both sides.
Step 1.5
Complete the square for .
Step 1.5.1
Use the form , to find the values of , , and .
Step 1.5.2
Consider the vertex form of a parabola.
Step 1.5.3
Find the value of using the formula .
Step 1.5.3.1
Substitute the values of and into the formula .
Step 1.5.3.2
Simplify the right side.
Step 1.5.3.2.1
Cancel the common factor of and .
Step 1.5.3.2.1.1
Factor out of .
Step 1.5.3.2.1.2
Move the negative one from the denominator of .
Step 1.5.3.2.2
Rewrite as .
Step 1.5.3.2.3
Multiply by .
Step 1.5.4
Find the value of using the formula .
Step 1.5.4.1
Substitute the values of , and into the formula .
Step 1.5.4.2
Simplify the right side.
Step 1.5.4.2.1
Simplify each term.
Step 1.5.4.2.1.1
Cancel the common factor of and .
Step 1.5.4.2.1.1.1
Rewrite as .
Step 1.5.4.2.1.1.2
Apply the product rule to .
Step 1.5.4.2.1.1.3
Raise to the power of .
Step 1.5.4.2.1.1.4
Multiply by .
Step 1.5.4.2.1.1.5
Factor out of .
Step 1.5.4.2.1.1.6
Move the negative one from the denominator of .
Step 1.5.4.2.1.2
Multiply .
Step 1.5.4.2.1.2.1
Multiply by .
Step 1.5.4.2.1.2.2
Multiply by .
Step 1.5.4.2.2
Add and .
Step 1.5.5
Substitute the values of , , and into the vertex form .
Step 1.6
Substitute for in the equation .
Step 1.7
Move to the right side of the equation by adding to both sides.
Step 1.8
Simplify .
Step 1.8.1
Add and .
Step 1.8.2
Subtract from .
Step 1.9
Divide each term by to make the right side equal to one.
Step 1.10
Simplify each term in the equation in order to set the right side equal to . The standard form of an ellipse or hyperbola requires the right side of the equation be .
Step 2
This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.
Step 3
Match the values in this hyperbola to those of the standard form. The variable represents the x-offset from the origin, represents the y-offset from origin, .
Step 4
The center of a hyperbola follows the form of . Substitute in the values of and .
Step 5
Step 5.1
Find the distance from the center to a focus of the hyperbola by using the following formula.
Step 5.2
Substitute the values of and in the formula.
Step 5.3
Simplify.
Step 5.3.1
Raise to the power of .
Step 5.3.2
Raise to the power of .
Step 5.3.3
Add and .
Step 5.3.4
Rewrite as .
Step 5.3.4.1
Factor out of .
Step 5.3.4.2
Rewrite as .
Step 5.3.5
Pull terms out from under the radical.
Step 6
Step 6.1
The first vertex of a hyperbola can be found by adding to .
Step 6.2
Substitute the known values of , , and into the formula and simplify.
Step 6.3
The second vertex of a hyperbola can be found by subtracting from .
Step 6.4
Substitute the known values of , , and into the formula and simplify.
Step 6.5
The vertices of a hyperbola follow the form of . Hyperbolas have two vertices.
Step 7
Step 7.1
The first focus of a hyperbola can be found by adding to .
Step 7.2
Substitute the known values of , , and into the formula and simplify.
Step 7.3
The second focus of a hyperbola can be found by subtracting from .
Step 7.4
Substitute the known values of , , and into the formula and simplify.
Step 7.5
The foci of a hyperbola follow the form of . Hyperbolas have two foci.
Step 8
Step 8.1
Find the eccentricity by using the following formula.
Step 8.2
Substitute the values of and into the formula.
Step 8.3
Simplify.
Step 8.3.1
Simplify the numerator.
Step 8.3.1.1
Raise to the power of .
Step 8.3.1.2
Raise to the power of .
Step 8.3.1.3
Add and .
Step 8.3.1.4
Rewrite as .
Step 8.3.1.4.1
Factor out of .
Step 8.3.1.4.2
Rewrite as .
Step 8.3.1.5
Pull terms out from under the radical.
Step 8.3.2
Cancel the common factor of .
Step 8.3.2.1
Cancel the common factor.
Step 8.3.2.2
Divide by .
Step 9
Step 9.1
Find the value of the focal parameter of the hyperbola by using the following formula.
Step 9.2
Substitute the values of and in the formula.
Step 9.3
Simplify.
Step 9.3.1
Raise to the power of .
Step 9.3.2
Cancel the common factor of and .
Step 9.3.2.1
Factor out of .
Step 9.3.2.2
Cancel the common factors.
Step 9.3.2.2.1
Factor out of .
Step 9.3.2.2.2
Cancel the common factor.
Step 9.3.2.2.3
Rewrite the expression.
Step 9.3.3
Multiply by .
Step 9.3.4
Combine and simplify the denominator.
Step 9.3.4.1
Multiply by .
Step 9.3.4.2
Raise to the power of .
Step 9.3.4.3
Raise to the power of .
Step 9.3.4.4
Use the power rule to combine exponents.
Step 9.3.4.5
Add and .
Step 9.3.4.6
Rewrite as .
Step 9.3.4.6.1
Use to rewrite as .
Step 9.3.4.6.2
Apply the power rule and multiply exponents, .
Step 9.3.4.6.3
Combine and .
Step 9.3.4.6.4
Cancel the common factor of .
Step 9.3.4.6.4.1
Cancel the common factor.
Step 9.3.4.6.4.2
Rewrite the expression.
Step 9.3.4.6.5
Evaluate the exponent.
Step 10
The asymptotes follow the form because this hyperbola opens up and down.
Step 11
Step 11.1
Remove parentheses.
Step 11.2
Simplify .
Step 11.2.1
Simplify each term.
Step 11.2.1.1
Multiply by .
Step 11.2.1.2
Apply the distributive property.
Step 11.2.1.3
Combine and .
Step 11.2.1.4
Cancel the common factor of .
Step 11.2.1.4.1
Cancel the common factor.
Step 11.2.1.4.2
Rewrite the expression.
Step 11.2.2
Add and .
Step 12
Step 12.1
Remove parentheses.
Step 12.2
Simplify .
Step 12.2.1
Simplify each term.
Step 12.2.1.1
Multiply by .
Step 12.2.1.2
Apply the distributive property.
Step 12.2.1.3
Combine and .
Step 12.2.1.4
Cancel the common factor of .
Step 12.2.1.4.1
Move the leading negative in into the numerator.
Step 12.2.1.4.2
Cancel the common factor.
Step 12.2.1.4.3
Rewrite the expression.
Step 12.2.2
Add and .
Step 13
This hyperbola has two asymptotes.
Step 14
These values represent the important values for graphing and analyzing a hyperbola.
Center:
Vertices:
Foci:
Eccentricity:
Focal Parameter:
Asymptotes: ,
Step 15