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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
is approximately which is positive so remove the absolute value
Step 3.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 3.1.5
Cancel the common factor of .
Step 3.1.5.1
Factor out of .
Step 3.1.5.2
Cancel the common factor.
Step 3.1.5.3
Rewrite the expression.
Step 3.1.6
Move to the left of .
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
is approximately which is positive so remove the absolute value
Step 3.2.4
Multiply the numerator by the reciprocal of the denominator.
Step 3.2.5
Cancel the common factor of .
Step 3.2.5.1
Factor out of .
Step 3.2.5.2
Cancel the common factor.
Step 3.2.5.3
Rewrite the expression.
Step 3.2.6
Move to the left of .
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
Multiply by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: None
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
Multiply by .
Step 6.1.2.2
Simplify each term.
Step 6.1.2.2.1
Cancel the common factor of and .
Step 6.1.2.2.1.1
Factor out of .
Step 6.1.2.2.1.2
Cancel the common factors.
Step 6.1.2.2.1.2.1
Factor out of .
Step 6.1.2.2.1.2.2
Cancel the common factor.
Step 6.1.2.2.1.2.3
Rewrite the expression.
Step 6.1.2.2.1.2.4
Divide by .
Step 6.1.2.2.2
The exact value of is .
Step 6.1.2.2.3
Multiply by .
Step 6.1.2.3
To write as a fraction with a common denominator, multiply by .
Step 6.1.2.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.1.2.4.1
Multiply by .
Step 6.1.2.4.2
Multiply by .
Step 6.1.2.5
Combine the numerators over the common denominator.
Step 6.1.2.6
Subtract from .
Step 6.1.2.7
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
Combine and .
Step 6.2.2.1.2
Multiply by .
Step 6.2.2.1.3
Reduce the expression by cancelling the common factors.
Step 6.2.2.1.3.1
Factor out of .
Step 6.2.2.1.3.2
Factor out of .
Step 6.2.2.1.3.3
Cancel the common factor.
Step 6.2.2.1.3.4
Rewrite the expression.
Step 6.2.2.1.4
Simplify the numerator.
Step 6.2.2.1.4.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.2.2.1.4.2
Cancel the common factor of .
Step 6.2.2.1.4.2.1
Factor out of .
Step 6.2.2.1.4.2.2
Cancel the common factor.
Step 6.2.2.1.4.2.3
Rewrite the expression.
Step 6.2.2.1.4.3
The exact value of is .
Step 6.2.2.1.5
Multiply by .
Step 6.2.2.1.6
Divide by .
Step 6.2.2.2
Subtract from .
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
Combine and .
Step 6.3.2.1.2
Multiply by .
Step 6.3.2.1.3
Reduce the expression by cancelling the common factors.
Step 6.3.2.1.3.1
Reduce the expression by cancelling the common factors.
Step 6.3.2.1.3.1.1
Factor out of .
Step 6.3.2.1.3.1.2
Factor out of .
Step 6.3.2.1.3.1.3
Cancel the common factor.
Step 6.3.2.1.3.1.4
Rewrite the expression.
Step 6.3.2.1.3.2
Divide by .
Step 6.3.2.1.4
Simplify the numerator.
Step 6.3.2.1.4.1
Cancel the common factor of .
Step 6.3.2.1.4.1.1
Cancel the common factor.
Step 6.3.2.1.4.1.2
Divide by .
Step 6.3.2.1.4.2
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 6.3.2.1.4.3
The exact value of is .
Step 6.3.2.1.4.4
Multiply by .
Step 6.3.2.1.5
Multiply by .
Step 6.3.2.1.6
Move the negative in front of the fraction.
Step 6.3.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.3.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.3.2.3.1
Multiply by .
Step 6.3.2.3.2
Multiply by .
Step 6.3.2.4
Combine the numerators over the common denominator.
Step 6.3.2.5
Subtract from .
Step 6.3.2.6
Move the negative in front of the fraction.
Step 6.3.2.7
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
Simplify each term.
Step 6.4.2.1.1
Combine and .
Step 6.4.2.1.2
Multiply by .
Step 6.4.2.1.3
Reduce the expression by cancelling the common factors.
Step 6.4.2.1.3.1
Factor out of .
Step 6.4.2.1.3.2
Factor out of .
Step 6.4.2.1.3.3
Cancel the common factor.
Step 6.4.2.1.3.4
Rewrite the expression.
Step 6.4.2.1.4
Simplify the numerator.
Step 6.4.2.1.4.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.4.2.1.4.2
Cancel the common factor of .
Step 6.4.2.1.4.2.1
Factor out of .
Step 6.4.2.1.4.2.2
Cancel the common factor.
Step 6.4.2.1.4.2.3
Rewrite the expression.
Step 6.4.2.1.4.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.4.2.1.4.4
The exact value of is .
Step 6.4.2.1.5
Multiply by .
Step 6.4.2.1.6
Divide by .
Step 6.4.2.2
Subtract from .
Step 6.4.2.3
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
Cancel the common factor of .
Step 6.5.2.1.1.1
Cancel the common factor.
Step 6.5.2.1.1.2
Divide by .
Step 6.5.2.1.2
Simplify the numerator.
Step 6.5.2.1.2.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.1.2.2
The exact value of is .
Step 6.5.2.1.3
Multiply by .
Step 6.5.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.5.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.5.2.3.1
Multiply by .
Step 6.5.2.3.2
Multiply by .
Step 6.5.2.4
Combine the numerators over the common denominator.
Step 6.5.2.5
Subtract from .
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: None
Vertical Shift:
Step 8