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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
Find the period of .
Step 3.1.1
The period of the function can be calculated using .
Step 3.1.2
Replace with in the formula for period.
Step 3.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.1.4
Cancel the common factor of and .
Step 3.1.4.1
Factor out of .
Step 3.1.4.2
Cancel the common factors.
Step 3.1.4.2.1
Factor out of .
Step 3.1.4.2.2
Cancel the common factor.
Step 3.1.4.2.3
Rewrite the expression.
Step 3.2
Find the period of .
Step 3.2.1
The period of the function can be calculated using .
Step 3.2.2
Replace with in the formula for period.
Step 3.2.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.4
Cancel the common factor of and .
Step 3.2.4.1
Factor out of .
Step 3.2.4.2
Cancel the common factors.
Step 3.2.4.2.1
Factor out of .
Step 3.2.4.2.2
Cancel the common factor.
Step 3.2.4.2.3
Rewrite the expression.
Step 3.3
The period of addition/subtraction of trig functions is the maximum of the individual periods.
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
Move the negative in front of the fraction.
Phase Shift:
Step 4.5
Multiply .
Step 4.5.1
Multiply by .
Phase Shift:
Step 4.5.2
Multiply by .
Phase Shift:
Step 4.5.3
Multiply by .
Phase Shift:
Step 4.5.4
Multiply by .
Phase Shift:
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the right)
Vertical Shift:
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
Simplify each term.
Step 6.1.2.1.1.1.1
Cancel the common factor of .
Step 6.1.2.1.1.1.1.1
Factor out of .
Step 6.1.2.1.1.1.1.2
Factor out of .
Step 6.1.2.1.1.1.1.3
Cancel the common factor.
Step 6.1.2.1.1.1.1.4
Rewrite the expression.
Step 6.1.2.1.1.1.2
Rewrite as .
Step 6.1.2.1.1.2
Combine the numerators over the common denominator.
Step 6.1.2.1.1.3
Add and .
Step 6.1.2.1.1.4
Divide by .
Step 6.1.2.1.1.5
The exact value of is .
Step 6.1.2.1.2
Multiply by .
Step 6.1.2.2
Simplify the expression.
Step 6.1.2.2.1
Write as a fraction with a common denominator.
Step 6.1.2.2.2
Combine the numerators over the common denominator.
Step 6.1.2.2.3
Add and .
Step 6.1.2.3
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
Simplify each term.
Step 6.2.2.1.1
Simplify the numerator.
Step 6.2.2.1.1.1
Simplify each term.
Step 6.2.2.1.1.1.1
Cancel the common factor of .
Step 6.2.2.1.1.1.1.1
Factor out of .
Step 6.2.2.1.1.1.1.2
Cancel the common factor.
Step 6.2.2.1.1.1.1.3
Rewrite the expression.
Step 6.2.2.1.1.1.2
Rewrite as .
Step 6.2.2.1.1.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.1.1.3
Combine and .
Step 6.2.2.1.1.4
Combine the numerators over the common denominator.
Step 6.2.2.1.1.5
Simplify the numerator.
Step 6.2.2.1.1.5.1
Multiply by .
Step 6.2.2.1.1.5.2
Add and .
Step 6.2.2.1.1.6
Move the negative in front of the fraction.
Step 6.2.2.1.1.7
Add full rotations of until the angle is greater than or equal to and less than .
Step 6.2.2.1.1.8
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.2.2.1.1.9
The exact value of is .
Step 6.2.2.1.2
Multiply by .
Step 6.2.2.1.3
Divide by .
Step 6.2.2.2
Add and .
Step 6.2.2.3
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
Simplify each term.
Step 6.3.2.1.1.1.1
Cancel the common factor of .
Step 6.3.2.1.1.1.1.1
Factor out of .
Step 6.3.2.1.1.1.1.2
Factor out of .
Step 6.3.2.1.1.1.1.3
Cancel the common factor.
Step 6.3.2.1.1.1.1.4
Rewrite the expression.
Step 6.3.2.1.1.1.2
Rewrite as .
Step 6.3.2.1.1.2
Combine the numerators over the common denominator.
Step 6.3.2.1.1.3
Add and .
Step 6.3.2.1.1.4
Cancel the common factor of and .
Step 6.3.2.1.1.4.1
Factor out of .
Step 6.3.2.1.1.4.2
Cancel the common factors.
Step 6.3.2.1.1.4.2.1
Factor out of .
Step 6.3.2.1.1.4.2.2
Cancel the common factor.
Step 6.3.2.1.1.4.2.3
Rewrite the expression.
Step 6.3.2.1.1.4.2.4
Divide by .
Step 6.3.2.1.1.5
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 6.3.2.1.1.6
The exact value of is .
Step 6.3.2.1.1.7
Multiply by .
Step 6.3.2.1.2
Multiply by .
Step 6.3.2.1.3
Move the negative in front of the fraction.
Step 6.3.2.2
Simplify the expression.
Step 6.3.2.2.1
Write as a fraction with a common denominator.
Step 6.3.2.2.2
Combine the numerators over the common denominator.
Step 6.3.2.2.3
Add and .
Step 6.3.2.2.4
Move the negative in front of the fraction.
Step 6.3.2.3
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
Add and .
Step 6.4.2.2
Simplify each term.
Step 6.4.2.2.1
Simplify the numerator.
Step 6.4.2.2.1.1
Combine and .
Step 6.4.2.2.1.2
Move the negative in front of the fraction.
Step 6.4.2.2.1.3
Add full rotations of until the angle is greater than or equal to and less than .
Step 6.4.2.2.1.4
The exact value of is .
Step 6.4.2.2.2
Multiply by .
Step 6.4.2.2.3
Divide by .
Step 6.4.2.3
Add and .
Step 6.4.2.4
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
Simplify each term.
Step 6.5.2.1.1.1.1
Cancel the common factor of .
Step 6.5.2.1.1.1.1.1
Factor out of .
Step 6.5.2.1.1.1.1.2
Factor out of .
Step 6.5.2.1.1.1.1.3
Cancel the common factor.
Step 6.5.2.1.1.1.1.4
Rewrite the expression.
Step 6.5.2.1.1.1.2
Rewrite as .
Step 6.5.2.1.1.2
Combine the numerators over the common denominator.
Step 6.5.2.1.1.3
Add and .
Step 6.5.2.1.1.4
Cancel the common factor of and .
Step 6.5.2.1.1.4.1
Factor out of .
Step 6.5.2.1.1.4.2
Cancel the common factors.
Step 6.5.2.1.1.4.2.1
Factor out of .
Step 6.5.2.1.1.4.2.2
Cancel the common factor.
Step 6.5.2.1.1.4.2.3
Rewrite the expression.
Step 6.5.2.1.1.4.2.4
Divide by .
Step 6.5.2.1.1.5
Add full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.1.1.6
The exact value of is .
Step 6.5.2.1.2
Multiply by .
Step 6.5.2.2
Simplify the expression.
Step 6.5.2.2.1
Write as a fraction with a common denominator.
Step 6.5.2.2.2
Combine the numerators over the common denominator.
Step 6.5.2.2.3
Add and .
Step 6.5.2.3
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 right)
Vertical Shift:
Step 8