Enter a problem...
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.4
Cancel the common factor of .
Step 3.4.1
Cancel the common factor.
Step 3.4.2
Divide 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
Simplify the numerator.
Step 4.3.1
To write as a fraction with a common denominator, multiply by .
Phase Shift:
Step 4.3.2
To write as a fraction with a common denominator, multiply by .
Phase Shift:
Step 4.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 4.3.3.1
Multiply by .
Phase Shift:
Step 4.3.3.2
Multiply by .
Phase Shift:
Step 4.3.3.3
Multiply by .
Phase Shift:
Step 4.3.3.4
Multiply by .
Phase Shift:
Phase Shift:
Step 4.3.4
Combine the numerators over the common denominator.
Phase Shift:
Step 4.3.5
Simplify the numerator.
Step 4.3.5.1
Move to the left of .
Phase Shift:
Step 4.3.5.2
Multiply by .
Phase Shift:
Phase Shift:
Phase Shift:
Step 4.4
Multiply the numerator by the reciprocal of the denominator.
Phase Shift:
Step 4.5
Multiply .
Step 4.5.1
Multiply by .
Phase Shift:
Step 4.5.2
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: 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
Cancel the common factor of .
Step 6.1.2.1.1.1
Move the leading negative in into the numerator.
Step 6.1.2.1.1.2
Factor out of .
Step 6.1.2.1.1.3
Cancel the common factor.
Step 6.1.2.1.1.4
Rewrite the expression.
Step 6.1.2.1.2
Move the negative in front of the fraction.
Step 6.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.1.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.1.2.3.1
Multiply by .
Step 6.1.2.3.2
Multiply by .
Step 6.1.2.4
Combine the numerators over the common denominator.
Step 6.1.2.5
Simplify each term.
Step 6.1.2.5.1
Simplify the numerator.
Step 6.1.2.5.1.1
Apply the distributive property.
Step 6.1.2.5.1.2
Multiply by .
Step 6.1.2.5.1.3
Multiply by .
Step 6.1.2.5.1.4
Move to the left of .
Step 6.1.2.5.1.5
Add and .
Step 6.1.2.5.1.6
Add and .
Step 6.1.2.5.2
Cancel the common factor of and .
Step 6.1.2.5.2.1
Factor out of .
Step 6.1.2.5.2.2
Cancel the common factors.
Step 6.1.2.5.2.2.1
Factor out of .
Step 6.1.2.5.2.2.2
Cancel the common factor.
Step 6.1.2.5.2.2.3
Rewrite the expression.
Step 6.1.2.6
Combine fractions.
Step 6.1.2.6.1
Combine the numerators over the common denominator.
Step 6.1.2.6.2
Simplify the expression.
Step 6.1.2.6.2.1
Subtract from .
Step 6.1.2.6.2.2
Divide by .
Step 6.1.2.7
The exact value of is .
Step 6.1.2.8
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 .
Step 6.2.2.1.1
Factor out of .
Step 6.2.2.1.2
Cancel the common factor.
Step 6.2.2.1.3
Rewrite the expression.
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
Add and .
Step 6.2.2.5.1
Reorder and .
Step 6.2.2.5.2
Add and .
Step 6.2.2.6
Cancel the common factor of and .
Step 6.2.2.6.1
Factor out of .
Step 6.2.2.6.2
Factor out of .
Step 6.2.2.6.3
Factor out of .
Step 6.2.2.6.4
Cancel the common factors.
Step 6.2.2.6.4.1
Factor out of .
Step 6.2.2.6.4.2
Cancel the common factor.
Step 6.2.2.6.4.3
Rewrite the expression.
Step 6.2.2.7
Combine the numerators over the common denominator.
Step 6.2.2.8
Simplify by subtracting numbers.
Step 6.2.2.8.1
Subtract from .
Step 6.2.2.8.2
Add and .
Step 6.2.2.9
The exact value of is .
Step 6.2.2.10
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
Cancel the common factor of .
Step 6.3.2.1.1
Factor out of .
Step 6.3.2.1.2
Cancel the common factor.
Step 6.3.2.1.3
Rewrite the expression.
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
Add and .
Step 6.3.2.5.1
Reorder and .
Step 6.3.2.5.2
Add and .
Step 6.3.2.6
Cancel the common factor of and .
Step 6.3.2.6.1
Factor out of .
Step 6.3.2.6.2
Factor out of .
Step 6.3.2.6.3
Factor out of .
Step 6.3.2.6.4
Cancel the common factors.
Step 6.3.2.6.4.1
Factor out of .
Step 6.3.2.6.4.2
Cancel the common factor.
Step 6.3.2.6.4.3
Rewrite the expression.
Step 6.3.2.7
Combine the numerators over the common denominator.
Step 6.3.2.8
Simplify by subtracting numbers.
Step 6.3.2.8.1
Subtract from .
Step 6.3.2.8.2
Add and .
Step 6.3.2.9
Cancel the common factor of .
Step 6.3.2.9.1
Cancel the common factor.
Step 6.3.2.9.2
Divide by .
Step 6.3.2.10
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 6.3.2.11
The exact value of is .
Step 6.3.2.12
Multiply by .
Step 6.3.2.13
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 .
Step 6.4.2.1.1
Factor out of .
Step 6.4.2.1.2
Cancel the common factor.
Step 6.4.2.1.3
Rewrite the expression.
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
Add and .
Step 6.4.2.5.1
Reorder and .
Step 6.4.2.5.2
Add and .
Step 6.4.2.6
Cancel the common factor of and .
Step 6.4.2.6.1
Factor out of .
Step 6.4.2.6.2
Factor out of .
Step 6.4.2.6.3
Factor out of .
Step 6.4.2.6.4
Cancel the common factors.
Step 6.4.2.6.4.1
Factor out of .
Step 6.4.2.6.4.2
Cancel the common factor.
Step 6.4.2.6.4.3
Rewrite the expression.
Step 6.4.2.7
Combine the numerators over the common denominator.
Step 6.4.2.8
Simplify by subtracting numbers.
Step 6.4.2.8.1
Subtract from .
Step 6.4.2.8.2
Add and .
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
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
Cancel the common factor of .
Step 6.5.2.1.1
Factor out of .
Step 6.5.2.1.2
Cancel the common factor.
Step 6.5.2.1.3
Rewrite the expression.
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
Add and .
Step 6.5.2.5.1
Reorder and .
Step 6.5.2.5.2
Add and .
Step 6.5.2.6
Cancel the common factor of and .
Step 6.5.2.6.1
Factor out of .
Step 6.5.2.6.2
Factor out of .
Step 6.5.2.6.3
Factor out of .
Step 6.5.2.6.4
Cancel the common factors.
Step 6.5.2.6.4.1
Factor out of .
Step 6.5.2.6.4.2
Cancel the common factor.
Step 6.5.2.6.4.3
Rewrite the expression.
Step 6.5.2.7
Combine the numerators over the common denominator.
Step 6.5.2.8
Simplify by subtracting numbers.
Step 6.5.2.8.1
Subtract from .
Step 6.5.2.8.2
Add and .
Step 6.5.2.9
Cancel the common factor of and .
Step 6.5.2.9.1
Factor out of .
Step 6.5.2.9.2
Cancel the common factors.
Step 6.5.2.9.2.1
Factor out of .
Step 6.5.2.9.2.2
Cancel the common factor.
Step 6.5.2.9.2.3
Rewrite the expression.
Step 6.5.2.9.2.4
Divide by .
Step 6.5.2.10
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.11
The exact value of is .
Step 6.5.2.12
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: None
Step 8