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Trigonometry Examples
Step 1
Step 1.1
Move all terms containing variables to the left side of the equation.
Step 1.1.1
Subtract from both sides of the equation.
Step 1.1.2
Reorder and .
Step 1.2
Flip the sign on each term of the equation so the term on the right side is positive.
Step 1.3
Divide each term by to make the right side equal to one.
Step 1.4
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
Rewrite as .
Step 5.3.1.1
Use to rewrite as .
Step 5.3.1.2
Apply the power rule and multiply exponents, .
Step 5.3.1.3
Combine and .
Step 5.3.1.4
Cancel the common factor of .
Step 5.3.1.4.1
Cancel the common factor.
Step 5.3.1.4.2
Rewrite the expression.
Step 5.3.1.5
Evaluate the exponent.
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
Simplify the expression.
Step 5.3.3.1
Add and .
Step 5.3.3.2
Rewrite as .
Step 5.3.4
Pull terms out from under the radical, assuming positive real numbers.
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
Rewrite as .
Step 8.3.1.1.1
Use to rewrite as .
Step 8.3.1.1.2
Apply the power rule and multiply exponents, .
Step 8.3.1.1.3
Combine and .
Step 8.3.1.1.4
Cancel the common factor of .
Step 8.3.1.1.4.1
Cancel the common factor.
Step 8.3.1.1.4.2
Rewrite the expression.
Step 8.3.1.1.5
Evaluate the exponent.
Step 8.3.1.2
Rewrite as .
Step 8.3.1.2.1
Use to rewrite as .
Step 8.3.1.2.2
Apply the power rule and multiply exponents, .
Step 8.3.1.2.3
Combine and .
Step 8.3.1.2.4
Cancel the common factor of .
Step 8.3.1.2.4.1
Cancel the common factor.
Step 8.3.1.2.4.2
Rewrite the expression.
Step 8.3.1.2.5
Evaluate the exponent.
Step 8.3.1.3
Add and .
Step 8.3.1.4
Rewrite as .
Step 8.3.1.5
Pull terms out from under the radical, assuming positive real numbers.
Step 8.3.2
Multiply by .
Step 8.3.3
Combine and simplify the denominator.
Step 8.3.3.1
Multiply by .
Step 8.3.3.2
Raise to the power of .
Step 8.3.3.3
Raise to the power of .
Step 8.3.3.4
Use the power rule to combine exponents.
Step 8.3.3.5
Add and .
Step 8.3.3.6
Rewrite as .
Step 8.3.3.6.1
Use to rewrite as .
Step 8.3.3.6.2
Apply the power rule and multiply exponents, .
Step 8.3.3.6.3
Combine and .
Step 8.3.3.6.4
Cancel the common factor of .
Step 8.3.3.6.4.1
Cancel the common factor.
Step 8.3.3.6.4.2
Rewrite the expression.
Step 8.3.3.6.5
Evaluate the exponent.
Step 8.3.4
Cancel the common factor of .
Step 8.3.4.1
Cancel the common factor.
Step 8.3.4.2
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
Rewrite as .
Step 9.3.1.1
Use to rewrite as .
Step 9.3.1.2
Apply the power rule and multiply exponents, .
Step 9.3.1.3
Combine and .
Step 9.3.1.4
Cancel the common factor of .
Step 9.3.1.4.1
Cancel the common factor.
Step 9.3.1.4.2
Rewrite the expression.
Step 9.3.1.5
Evaluate the exponent.
Step 9.3.2
Divide by .
Step 10
The asymptotes follow the form because this hyperbola opens left and right.
Step 11
Step 11.1
Add and .
Step 11.2
Multiply by .
Step 12
Step 12.1
Add and .
Step 12.2
Rewrite as .
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