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Trigonometry Examples
Step 1
Rewrite the expression as .
Step 2
Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.
Step 3
Find the amplitude .
Amplitude:
Step 4
Step 4.1
Find the period of .
Step 4.1.1
The period of the function can be calculated using .
Step 4.1.2
Replace with in the formula for period.
Step 4.1.3
is approximately which is positive so remove the absolute value
Step 4.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.1.5
Multiply by .
Step 4.2
Find the period of .
Step 4.2.1
The period of the function can be calculated using .
Step 4.2.2
Replace with in the formula for period.
Step 4.2.3
is approximately which is positive so remove the absolute value
Step 4.2.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.5
Multiply by .
Step 4.3
The period of addition/subtraction of trig functions is the maximum of the individual periods.
Step 5
Step 5.1
The phase shift of the function can be calculated from .
Phase Shift:
Step 5.2
Replace the values of and in the equation for phase shift.
Phase Shift:
Step 5.3
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Step 5.4
Multiply by .
Phase Shift:
Phase Shift:
Step 6
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the left)
Vertical Shift:
Step 7
Step 7.1
Find the point at .
Step 7.1.1
Replace the variable with in the expression.
Step 7.1.2
Simplify the result.
Step 7.1.2.1
To write as a fraction with a common denominator, multiply by .
Step 7.1.2.2
Combine and .
Step 7.1.2.3
Combine the numerators over the common denominator.
Step 7.1.2.4
Simplify the numerator.
Step 7.1.2.4.1
Multiply by .
Step 7.1.2.4.2
Add and .
Step 7.1.2.5
Simplify each term.
Step 7.1.2.5.1
Cancel the common factor of and .
Step 7.1.2.5.1.1
Factor out of .
Step 7.1.2.5.1.2
Cancel the common factors.
Step 7.1.2.5.1.2.1
Factor out of .
Step 7.1.2.5.1.2.2
Cancel the common factor.
Step 7.1.2.5.1.2.3
Rewrite the expression.
Step 7.1.2.5.1.2.4
Divide by .
Step 7.1.2.5.2
The exact value of is .
Step 7.1.2.5.3
Multiply by .
Step 7.1.2.6
To write as a fraction with a common denominator, multiply by .
Step 7.1.2.7
Combine and .
Step 7.1.2.8
Combine the numerators over the common denominator.
Step 7.1.2.9
Simplify the numerator.
Step 7.1.2.9.1
Multiply by .
Step 7.1.2.9.2
Add and .
Step 7.1.2.10
The final answer is .
Step 7.2
Find the point at .
Step 7.2.1
Replace the variable with in the expression.
Step 7.2.2
Simplify the result.
Step 7.2.2.1
To write as a fraction with a common denominator, multiply by .
Step 7.2.2.2
Combine fractions.
Step 7.2.2.2.1
Combine and .
Step 7.2.2.2.2
Simplify the expression.
Step 7.2.2.2.2.1
Combine the numerators over the common denominator.
Step 7.2.2.2.2.2
Multiply by .
Step 7.2.2.2.2.3
Add and .
Step 7.2.2.2.2.4
Add and .
Step 7.2.2.3
Simplify each term.
Step 7.2.2.3.1
Cancel the common factor of and .
Step 7.2.2.3.1.1
Factor out of .
Step 7.2.2.3.1.2
Cancel the common factors.
Step 7.2.2.3.1.2.1
Factor out of .
Step 7.2.2.3.1.2.2
Cancel the common factor.
Step 7.2.2.3.1.2.3
Rewrite the expression.
Step 7.2.2.3.2
The exact value of is .
Step 7.2.2.3.3
Multiply by .
Step 7.2.2.3.4
Divide by .
Step 7.2.2.4
Add and .
Step 7.2.2.5
The final answer is .
Step 7.3
Find the point at .
Step 7.3.1
Replace the variable with in the expression.
Step 7.3.2
Simplify the result.
Step 7.3.2.1
To write as a fraction with a common denominator, multiply by .
Step 7.3.2.2
Combine fractions.
Step 7.3.2.2.1
Combine and .
Step 7.3.2.2.2
Simplify the expression.
Step 7.3.2.2.2.1
Combine the numerators over the common denominator.
Step 7.3.2.2.2.2
Multiply by .
Step 7.3.2.2.2.3
Add and .
Step 7.3.2.2.2.4
Add and .
Step 7.3.2.3
Simplify each term.
Step 7.3.2.3.1
Cancel the common factor of .
Step 7.3.2.3.1.1
Cancel the common factor.
Step 7.3.2.3.1.2
Divide by .
Step 7.3.2.3.2
Simplify the numerator.
Step 7.3.2.3.2.1
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 7.3.2.3.2.2
The exact value of is .
Step 7.3.2.3.2.3
Multiply by .
Step 7.3.2.3.3
Multiply by .
Step 7.3.2.3.4
Move the negative in front of the fraction.
Step 7.3.2.4
To write as a fraction with a common denominator, multiply by .
Step 7.3.2.5
Combine and .
Step 7.3.2.6
Combine the numerators over the common denominator.
Step 7.3.2.7
Simplify the numerator.
Step 7.3.2.7.1
Multiply by .
Step 7.3.2.7.2
Subtract from .
Step 7.3.2.8
The final answer is .
Step 7.4
Find the point at .
Step 7.4.1
Replace the variable with in the expression.
Step 7.4.2
Simplify the result.
Step 7.4.2.1
To write as a fraction with a common denominator, multiply by .
Step 7.4.2.2
Combine fractions.
Step 7.4.2.2.1
Combine and .
Step 7.4.2.2.2
Simplify the expression.
Step 7.4.2.2.2.1
Combine the numerators over the common denominator.
Step 7.4.2.2.2.2
Multiply by .
Step 7.4.2.2.2.3
Add and .
Step 7.4.2.2.2.4
Add and .
Step 7.4.2.3
Simplify each term.
Step 7.4.2.3.1
Cancel the common factor of and .
Step 7.4.2.3.1.1
Factor out of .
Step 7.4.2.3.1.2
Cancel the common factors.
Step 7.4.2.3.1.2.1
Factor out of .
Step 7.4.2.3.1.2.2
Cancel the common factor.
Step 7.4.2.3.1.2.3
Rewrite the expression.
Step 7.4.2.3.2
Simplify the numerator.
Step 7.4.2.3.2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 7.4.2.3.2.2
The exact value of is .
Step 7.4.2.3.3
Multiply by .
Step 7.4.2.3.4
Divide by .
Step 7.4.2.4
Add and .
Step 7.4.2.5
The final answer is .
Step 7.5
Find the point at .
Step 7.5.1
Replace the variable with in the expression.
Step 7.5.2
Simplify the result.
Step 7.5.2.1
To write as a fraction with a common denominator, multiply by .
Step 7.5.2.2
Combine fractions.
Step 7.5.2.2.1
Combine and .
Step 7.5.2.2.2
Simplify the expression.
Step 7.5.2.2.2.1
Combine the numerators over the common denominator.
Step 7.5.2.2.2.2
Multiply by .
Step 7.5.2.2.2.3
Add and .
Step 7.5.2.2.2.4
Add and .
Step 7.5.2.3
Simplify each term.
Step 7.5.2.3.1
Cancel the common factor of and .
Step 7.5.2.3.1.1
Factor out of .
Step 7.5.2.3.1.2
Cancel the common factors.
Step 7.5.2.3.1.2.1
Factor out of .
Step 7.5.2.3.1.2.2
Cancel the common factor.
Step 7.5.2.3.1.2.3
Rewrite the expression.
Step 7.5.2.3.1.2.4
Divide by .
Step 7.5.2.3.2
Simplify the numerator.
Step 7.5.2.3.2.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 7.5.2.3.2.2
The exact value of is .
Step 7.5.2.3.3
Multiply by .
Step 7.5.2.4
To write as a fraction with a common denominator, multiply by .
Step 7.5.2.5
Combine and .
Step 7.5.2.6
Combine the numerators over the common denominator.
Step 7.5.2.7
Simplify the numerator.
Step 7.5.2.7.1
Multiply by .
Step 7.5.2.7.2
Add and .
Step 7.5.2.8
The final answer is .
Step 7.6
List the points in a table.
Step 8
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:
Step 9