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Trigonometry Examples
Step 1
Step 1.1
Add to 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
Move the negative in front of the fraction.
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
Cancel the common factor of and .
Step 1.2.4.2.1.3.1
Factor out of .
Step 1.2.4.2.1.3.2
Cancel the common factors.
Step 1.2.4.2.1.3.2.1
Factor out of .
Step 1.2.4.2.1.3.2.2
Cancel the common factor.
Step 1.2.4.2.1.3.2.3
Rewrite the expression.
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
Multiply by .
Step 1.5.3.2.2
Move the negative in front of the fraction.
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
Raise to the power of .
Step 1.5.4.2.1.2
Multiply by .
Step 1.5.4.2.1.3
Move the negative in front of the fraction.
Step 1.5.4.2.1.4
Multiply .
Step 1.5.4.2.1.4.1
Multiply by .
Step 1.5.4.2.1.4.2
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
Find the common denominator.
Step 1.8.1.1
Write as a fraction with denominator .
Step 1.8.1.2
Multiply by .
Step 1.8.1.3
Multiply by .
Step 1.8.1.4
Multiply by .
Step 1.8.1.5
Multiply by .
Step 1.8.1.6
Multiply by .
Step 1.8.2
Combine the numerators over the common denominator.
Step 1.8.3
Simplify the expression.
Step 1.8.3.1
Multiply by .
Step 1.8.3.2
Add and .
Step 1.8.3.3
Subtract from .
Step 1.8.4
Cancel the common factor of and .
Step 1.8.4.1
Factor out of .
Step 1.8.4.2
Cancel the common factors.
Step 1.8.4.2.1
Factor out of .
Step 1.8.4.2.2
Cancel the common factor.
Step 1.8.4.2.3
Rewrite the expression.
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
Apply the product rule to .
Step 5.3.2
Rewrite as .
Step 5.3.2.1
Use to rewrite as .
Step 5.3.2.2
Apply the power rule and multiply exponents, .
Step 5.3.2.3
Combine and .
Step 5.3.2.4
Cancel the common factor of .
Step 5.3.2.4.1
Cancel the common factor.
Step 5.3.2.4.2
Rewrite the expression.
Step 5.3.2.5
Evaluate the exponent.
Step 5.3.3
Raise to the power of .
Step 5.3.4
Cancel the common factor of and .
Step 5.3.4.1
Factor out of .
Step 5.3.4.2
Cancel the common factors.
Step 5.3.4.2.1
Factor out of .
Step 5.3.4.2.2
Cancel the common factor.
Step 5.3.4.2.3
Rewrite the expression.
Step 5.3.5
Apply the product rule to .
Step 5.3.6
Rewrite as .
Step 5.3.6.1
Use to rewrite as .
Step 5.3.6.2
Apply the power rule and multiply exponents, .
Step 5.3.6.3
Combine and .
Step 5.3.6.4
Cancel the common factor of .
Step 5.3.6.4.1
Cancel the common factor.
Step 5.3.6.4.2
Rewrite the expression.
Step 5.3.6.5
Evaluate the exponent.
Step 5.3.7
Raise to the power of .
Step 5.3.8
To write as a fraction with a common denominator, multiply by .
Step 5.3.9
To write as a fraction with a common denominator, multiply by .
Step 5.3.10
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 5.3.10.1
Multiply by .
Step 5.3.10.2
Multiply by .
Step 5.3.10.3
Multiply by .
Step 5.3.10.4
Multiply by .
Step 5.3.11
Combine the numerators over the common denominator.
Step 5.3.12
Simplify the numerator.
Step 5.3.12.1
Multiply by .
Step 5.3.12.2
Multiply by .
Step 5.3.12.3
Add and .
Step 5.3.13
Rewrite as .
Step 5.3.14
Simplify the numerator.
Step 5.3.14.1
Rewrite as .
Step 5.3.14.1.1
Factor out of .
Step 5.3.14.1.2
Rewrite as .
Step 5.3.14.2
Pull terms out from under the radical.
Step 5.3.15
Simplify the denominator.
Step 5.3.15.1
Rewrite as .
Step 5.3.15.1.1
Factor out of .
Step 5.3.15.1.2
Rewrite as .
Step 5.3.15.2
Pull terms out from under the radical.
Step 5.3.16
Multiply by .
Step 5.3.17
Combine and simplify the denominator.
Step 5.3.17.1
Multiply by .
Step 5.3.17.2
Move .
Step 5.3.17.3
Raise to the power of .
Step 5.3.17.4
Raise to the power of .
Step 5.3.17.5
Use the power rule to combine exponents.
Step 5.3.17.6
Add and .
Step 5.3.17.7
Rewrite as .
Step 5.3.17.7.1
Use to rewrite as .
Step 5.3.17.7.2
Apply the power rule and multiply exponents, .
Step 5.3.17.7.3
Combine and .
Step 5.3.17.7.4
Cancel the common factor of .
Step 5.3.17.7.4.1
Cancel the common factor.
Step 5.3.17.7.4.2
Rewrite the expression.
Step 5.3.17.7.5
Evaluate the exponent.
Step 5.3.18
Simplify the numerator.
Step 5.3.18.1
Combine using the product rule for radicals.
Step 5.3.18.2
Multiply by .
Step 5.3.19
Multiply by .
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
Multiply the numerator by the reciprocal of the denominator.
Step 8.3.2
Apply the product rule to .
Step 8.3.3
Rewrite as .
Step 8.3.3.1
Use to rewrite as .
Step 8.3.3.2
Apply the power rule and multiply exponents, .
Step 8.3.3.3
Combine and .
Step 8.3.3.4
Cancel the common factor of .
Step 8.3.3.4.1
Cancel the common factor.
Step 8.3.3.4.2
Rewrite the expression.
Step 8.3.3.5
Evaluate the exponent.
Step 8.3.4
Raise to the power of .
Step 8.3.5
Cancel the common factor of and .
Step 8.3.5.1
Factor out of .
Step 8.3.5.2
Cancel the common factors.
Step 8.3.5.2.1
Factor out of .
Step 8.3.5.2.2
Cancel the common factor.
Step 8.3.5.2.3
Rewrite the expression.
Step 8.3.6
Apply the product rule to .
Step 8.3.7
Rewrite as .
Step 8.3.7.1
Use to rewrite as .
Step 8.3.7.2
Apply the power rule and multiply exponents, .
Step 8.3.7.3
Combine and .
Step 8.3.7.4
Cancel the common factor of .
Step 8.3.7.4.1
Cancel the common factor.
Step 8.3.7.4.2
Rewrite the expression.
Step 8.3.7.5
Evaluate the exponent.
Step 8.3.8
Raise to the power of .
Step 8.3.9
To write as a fraction with a common denominator, multiply by .
Step 8.3.10
To write as a fraction with a common denominator, multiply by .
Step 8.3.11
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 8.3.11.1
Multiply by .
Step 8.3.11.2
Multiply by .
Step 8.3.11.3
Multiply by .
Step 8.3.11.4
Multiply by .
Step 8.3.12
Combine the numerators over the common denominator.
Step 8.3.13
Simplify the numerator.
Step 8.3.13.1
Multiply by .
Step 8.3.13.2
Multiply by .
Step 8.3.13.3
Add and .
Step 8.3.14
Rewrite as .
Step 8.3.15
Simplify the numerator.
Step 8.3.15.1
Rewrite as .
Step 8.3.15.1.1
Factor out of .
Step 8.3.15.1.2
Rewrite as .
Step 8.3.15.2
Pull terms out from under the radical.
Step 8.3.16
Simplify the denominator.
Step 8.3.16.1
Rewrite as .
Step 8.3.16.1.1
Factor out of .
Step 8.3.16.1.2
Rewrite as .
Step 8.3.16.2
Pull terms out from under the radical.
Step 8.3.17
Simplify terms.
Step 8.3.17.1
Cancel the common factor of .
Step 8.3.17.1.1
Factor out of .
Step 8.3.17.1.2
Factor out of .
Step 8.3.17.1.3
Cancel the common factor.
Step 8.3.17.1.4
Rewrite the expression.
Step 8.3.17.2
Multiply by .
Step 8.3.17.3
Multiply by .
Step 8.3.17.4
Combine using the product rule for radicals.
Step 8.3.17.5
Multiply by .
Step 8.3.18
Combine and into a single radical.
Step 8.3.19
Cancel the common factor of and .
Step 8.3.19.1
Factor out of .
Step 8.3.19.2
Cancel the common factors.
Step 8.3.19.2.1
Factor out of .
Step 8.3.19.2.2
Cancel the common factor.
Step 8.3.19.2.3
Rewrite the expression.
Step 8.3.20
Rewrite as .
Step 8.3.21
Any root of is .
Step 8.3.22
Simplify the denominator.
Step 8.3.22.1
Rewrite as .
Step 8.3.22.1.1
Factor out of .
Step 8.3.22.1.2
Rewrite as .
Step 8.3.22.2
Pull terms out from under the radical.
Step 8.3.23
Simplify terms.
Step 8.3.23.1
Cancel the common factor of .
Step 8.3.23.1.1
Factor out of .
Step 8.3.23.1.2
Factor out of .
Step 8.3.23.1.3
Cancel the common factor.
Step 8.3.23.1.4
Rewrite the expression.
Step 8.3.23.2
Combine and .
Step 8.3.24
Multiply by .
Step 8.3.25
Combine and simplify the denominator.
Step 8.3.25.1
Multiply by .
Step 8.3.25.2
Raise to the power of .
Step 8.3.25.3
Raise to the power of .
Step 8.3.25.4
Use the power rule to combine exponents.
Step 8.3.25.5
Add and .
Step 8.3.25.6
Rewrite as .
Step 8.3.25.6.1
Use to rewrite as .
Step 8.3.25.6.2
Apply the power rule and multiply exponents, .
Step 8.3.25.6.3
Combine and .
Step 8.3.25.6.4
Cancel the common factor of .
Step 8.3.25.6.4.1
Cancel the common factor.
Step 8.3.25.6.4.2
Rewrite the expression.
Step 8.3.25.6.5
Evaluate the exponent.
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
Multiply the numerator by the reciprocal of the denominator.
Step 9.3.2
Combine.
Step 9.3.3
Simplify the expression.
Step 9.3.3.1
Raise to the power of .
Step 9.3.3.2
Multiply by .
Step 9.3.3.3
Multiply by .
Step 9.3.4
Multiply the numerator by the reciprocal of the denominator.
Step 9.3.5
Cancel the common factor of .
Step 9.3.5.1
Factor out of .
Step 9.3.5.2
Cancel the common factor.
Step 9.3.5.3
Rewrite the expression.
Step 9.3.6
Multiply by .
Step 9.3.7
Multiply by .
Step 10
The asymptotes follow the form because this hyperbola opens left and right.
Step 11
Step 11.1
Multiply by .
Step 11.2
Remove parentheses.
Step 11.3
Simplify each term.
Step 11.3.1
Apply the distributive property.
Step 11.3.2
Combine and .
Step 11.3.3
Cancel the common factor of .
Step 11.3.3.1
Move the leading negative in into the numerator.
Step 11.3.3.2
Factor out of .
Step 11.3.3.3
Factor out of .
Step 11.3.3.4
Cancel the common factor.
Step 11.3.3.5
Rewrite the expression.
Step 11.3.4
Multiply by .
Step 11.3.5
Multiply by .
Step 11.3.6
Simplify each term.
Step 11.3.6.1
Simplify the numerator.
Step 11.3.6.1.1
Move to the left of .
Step 11.3.6.1.2
Rewrite as .
Step 11.3.6.2
Move the negative in front of the fraction.
Step 12
Step 12.1
Multiply by .
Step 12.2
Remove parentheses.
Step 12.3
Simplify each term.
Step 12.3.1
Apply the distributive property.
Step 12.3.2
Combine and .
Step 12.3.3
Cancel the common factor of .
Step 12.3.3.1
Move the leading negative in into the numerator.
Step 12.3.3.2
Move the leading negative in into the numerator.
Step 12.3.3.3
Factor out of .
Step 12.3.3.4
Factor out of .
Step 12.3.3.5
Cancel the common factor.
Step 12.3.3.6
Rewrite the expression.
Step 12.3.4
Multiply by .
Step 12.3.5
Multiply by .
Step 12.3.6
Multiply by .
Step 12.3.7
Multiply by .
Step 12.3.8
Move to the left of .
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