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Trigonometry Examples
Step 1
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Step 2
Find the amplitude .
Amplitude:
Step 3
Step 3.1
The period of the function can be calculated using .
Step 3.2
Replace with in the formula for period.
Step 3.3
is approximately which is positive so remove the absolute value
Step 3.4
Multiply the numerator by the reciprocal of the denominator.
Step 3.5
Multiply by .
Step 4
Step 4.1
The phase shift of the function can be calculated from .
Phase Shift:
Step 4.2
Replace the values of and in the equation for phase shift.
Phase Shift:
Step 4.3
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Step 4.4
Apply the distributive property.
Phase Shift:
Step 4.5
Multiply .
Step 4.5.1
Multiply by .
Phase Shift:
Step 4.5.2
Multiply by .
Phase Shift:
Phase Shift:
Step 4.6
Multiply by .
Phase Shift:
Step 4.7
Apply the distributive property.
Phase Shift:
Step 4.8
Cancel the common factor of .
Step 4.8.1
Factor out of .
Phase Shift:
Step 4.8.2
Cancel the common factor.
Phase Shift:
Step 4.8.3
Rewrite the expression.
Phase Shift:
Phase Shift:
Step 4.9
Multiply by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift: None
Step 6
Step 6.1
Find the point at .
Step 6.1.1
Replace the variable with in the expression.
Step 6.1.2
Simplify the result.
Step 6.1.2.1
Simplify each term.
Step 6.1.2.1.1
Simplify the numerator.
Step 6.1.2.1.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.1.2.1.1.2
Combine and .
Step 6.1.2.1.1.3
Combine the numerators over the common denominator.
Step 6.1.2.1.1.4
Multiply by .
Step 6.1.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 6.1.2.1.3
Multiply .
Step 6.1.2.1.3.1
Multiply by .
Step 6.1.2.1.3.2
Multiply by .
Step 6.1.2.2
Simplify terms.
Step 6.1.2.2.1
Combine the numerators over the common denominator.
Step 6.1.2.2.2
Subtract from .
Step 6.1.2.2.3
Simplify the expression.
Step 6.1.2.2.3.1
Subtract from .
Step 6.1.2.2.3.2
Divide by .
Step 6.1.2.2.3.3
Subtract from .
Step 6.1.2.3
The exact value of is .
Step 6.1.2.4
Multiply by .
Step 6.1.2.5
The final answer is .
Step 6.2
Find the point at .
Step 6.2.1
Replace the variable with in the expression.
Step 6.2.2
Simplify the result.
Step 6.2.2.1
Cancel the common factor of and .
Step 6.2.2.1.1
Factor out of .
Step 6.2.2.1.2
Factor out of .
Step 6.2.2.1.3
Factor out of .
Step 6.2.2.1.4
Cancel the common factors.
Step 6.2.2.1.4.1
Factor out of .
Step 6.2.2.1.4.2
Cancel the common factor.
Step 6.2.2.1.4.3
Rewrite the expression.
Step 6.2.2.1.4.4
Divide by .
Step 6.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.2.2.3.1
Multiply by .
Step 6.2.2.3.2
Multiply by .
Step 6.2.2.4
Combine the numerators over the common denominator.
Step 6.2.2.5
Find the common denominator.
Step 6.2.2.5.1
Write as a fraction with denominator .
Step 6.2.2.5.2
Multiply by .
Step 6.2.2.5.3
Multiply by .
Step 6.2.2.5.4
Write as a fraction with denominator .
Step 6.2.2.5.5
Multiply by .
Step 6.2.2.5.6
Multiply by .
Step 6.2.2.6
Combine the numerators over the common denominator.
Step 6.2.2.7
Simplify each term.
Step 6.2.2.7.1
Multiply by .
Step 6.2.2.7.2
Multiply by .
Step 6.2.2.7.3
Multiply by .
Step 6.2.2.8
Simplify terms.
Step 6.2.2.8.1
Subtract from .
Step 6.2.2.8.2
Simplify by adding numbers.
Step 6.2.2.8.2.1
Add and .
Step 6.2.2.8.2.2
Add and .
Step 6.2.2.8.3
Cancel the common factor of and .
Step 6.2.2.8.3.1
Factor out of .
Step 6.2.2.8.3.2
Cancel the common factors.
Step 6.2.2.8.3.2.1
Factor out of .
Step 6.2.2.8.3.2.2
Cancel the common factor.
Step 6.2.2.8.3.2.3
Rewrite the expression.
Step 6.2.2.9
The exact value of is .
Step 6.2.2.10
Multiply by .
Step 6.2.2.11
The final answer is .
Step 6.3
Find the point at .
Step 6.3.1
Replace the variable with in the expression.
Step 6.3.2
Simplify the result.
Step 6.3.2.1
Simplify each term.
Step 6.3.2.1.1
Simplify the numerator.
Step 6.3.2.1.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.3.2.1.1.2
Combine and .
Step 6.3.2.1.1.3
Combine the numerators over the common denominator.
Step 6.3.2.1.1.4
Multiply by .
Step 6.3.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 6.3.2.1.3
Multiply .
Step 6.3.2.1.3.1
Multiply by .
Step 6.3.2.1.3.2
Multiply by .
Step 6.3.2.2
Simplify terms.
Step 6.3.2.2.1
Combine the numerators over the common denominator.
Step 6.3.2.2.2
Subtract from .
Step 6.3.2.2.3
Cancel the common factor of and .
Step 6.3.2.2.3.1
Factor out of .
Step 6.3.2.2.3.2
Factor out of .
Step 6.3.2.2.3.3
Factor out of .
Step 6.3.2.2.3.4
Cancel the common factors.
Step 6.3.2.2.3.4.1
Factor out of .
Step 6.3.2.2.3.4.2
Cancel the common factor.
Step 6.3.2.2.3.4.3
Rewrite the expression.
Step 6.3.2.2.3.4.4
Divide by .
Step 6.3.2.2.4
Simplify by subtracting numbers.
Step 6.3.2.2.4.1
Subtract from .
Step 6.3.2.2.4.2
Add and .
Step 6.3.2.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 6.3.2.4
The exact value of is .
Step 6.3.2.5
Multiply .
Step 6.3.2.5.1
Multiply by .
Step 6.3.2.5.2
Multiply by .
Step 6.3.2.6
The final answer is .
Step 6.4
Find the point at .
Step 6.4.1
Replace the variable with in the expression.
Step 6.4.2
Simplify the result.
Step 6.4.2.1
Cancel the common factor of and .
Step 6.4.2.1.1
Factor out of .
Step 6.4.2.1.2
Factor out of .
Step 6.4.2.1.3
Factor out of .
Step 6.4.2.1.4
Cancel the common factors.
Step 6.4.2.1.4.1
Factor out of .
Step 6.4.2.1.4.2
Cancel the common factor.
Step 6.4.2.1.4.3
Rewrite the expression.
Step 6.4.2.1.4.4
Divide by .
Step 6.4.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.4.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.4.2.3.1
Multiply by .
Step 6.4.2.3.2
Multiply by .
Step 6.4.2.4
Combine the numerators over the common denominator.
Step 6.4.2.5
Find the common denominator.
Step 6.4.2.5.1
Write as a fraction with denominator .
Step 6.4.2.5.2
Multiply by .
Step 6.4.2.5.3
Multiply by .
Step 6.4.2.5.4
Write as a fraction with denominator .
Step 6.4.2.5.5
Multiply by .
Step 6.4.2.5.6
Multiply by .
Step 6.4.2.6
Combine the numerators over the common denominator.
Step 6.4.2.7
Simplify each term.
Step 6.4.2.7.1
Multiply by .
Step 6.4.2.7.2
Multiply by .
Step 6.4.2.7.3
Multiply by .
Step 6.4.2.8
Simplify terms.
Step 6.4.2.8.1
Subtract from .
Step 6.4.2.8.2
Simplify by adding numbers.
Step 6.4.2.8.2.1
Add and .
Step 6.4.2.8.2.2
Add and .
Step 6.4.2.8.3
Cancel the common factor of and .
Step 6.4.2.8.3.1
Factor out of .
Step 6.4.2.8.3.2
Cancel the common factors.
Step 6.4.2.8.3.2.1
Factor out of .
Step 6.4.2.8.3.2.2
Cancel the common factor.
Step 6.4.2.8.3.2.3
Rewrite the expression.
Step 6.4.2.9
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.4.2.10
The exact value of is .
Step 6.4.2.11
Multiply by .
Step 6.4.2.12
The final answer is .
Step 6.5
Find the point at .
Step 6.5.1
Replace the variable with in the expression.
Step 6.5.2
Simplify the result.
Step 6.5.2.1
Simplify each term.
Step 6.5.2.1.1
Simplify the numerator.
Step 6.5.2.1.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.5.2.1.1.2
Combine and .
Step 6.5.2.1.1.3
Combine the numerators over the common denominator.
Step 6.5.2.1.1.4
Multiply by .
Step 6.5.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 6.5.2.1.3
Multiply .
Step 6.5.2.1.3.1
Multiply by .
Step 6.5.2.1.3.2
Multiply by .
Step 6.5.2.2
Simplify terms.
Step 6.5.2.2.1
Combine the numerators over the common denominator.
Step 6.5.2.2.2
Subtract from .
Step 6.5.2.2.3
Cancel the common factor of and .
Step 6.5.2.2.3.1
Factor out of .
Step 6.5.2.2.3.2
Factor out of .
Step 6.5.2.2.3.3
Factor out of .
Step 6.5.2.2.3.4
Cancel the common factors.
Step 6.5.2.2.3.4.1
Factor out of .
Step 6.5.2.2.3.4.2
Cancel the common factor.
Step 6.5.2.2.3.4.3
Rewrite the expression.
Step 6.5.2.2.3.4.4
Divide by .
Step 6.5.2.2.4
Simplify by subtracting numbers.
Step 6.5.2.2.4.1
Subtract from .
Step 6.5.2.2.4.2
Add and .
Step 6.5.2.3
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.4
The exact value of is .
Step 6.5.2.5
Multiply by .
Step 6.5.2.6
The final answer is .
Step 6.6
List the points in a table.
Step 7
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift: None
Step 8