Trigonometry Examples

Graph y=cos(1/2x)
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
Find the period of .
Tap for more steps...
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
Find the phase shift using the formula .
Tap for more steps...
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: None
Step 6
Select a few points to graph.
Tap for more steps...
Step 6.1
Find the point at .
Tap for more steps...
Step 6.1.1
Replace the variable with in the expression.
Step 6.1.2
Simplify the result.
Tap for more steps...
Step 6.1.2.1
Divide by .
Step 6.1.2.2
The exact value of is .
Step 6.1.2.3
The final answer is .
Step 6.2
Find the point at .
Tap for more steps...
Step 6.2.1
Replace the variable with in the expression.
Step 6.2.2
Simplify the result.
Tap for more steps...
Step 6.2.2.1
The exact value of is .
Step 6.2.2.2
The final answer is .
Step 6.3
Find the point at .
Tap for more steps...
Step 6.3.1
Replace the variable with in the expression.
Step 6.3.2
Simplify the result.
Tap for more steps...
Step 6.3.2.1
Cancel the common factor of .
Tap for more steps...
Step 6.3.2.1.1
Cancel the common factor.
Step 6.3.2.1.2
Divide by .
Step 6.3.2.2
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 6.3.2.3
The exact value of is .
Step 6.3.2.4
Multiply by .
Step 6.3.2.5
The final answer is .
Step 6.4
Find the point at .
Tap for more steps...
Step 6.4.1
Replace the variable with in the expression.
Step 6.4.2
Simplify the result.
Tap for more steps...
Step 6.4.2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 6.4.2.2
The exact value of is .
Step 6.4.2.3
The final answer is .
Step 6.5
Find the point at .
Tap for more steps...
Step 6.5.1
Replace the variable with in the expression.
Step 6.5.2
Simplify the result.
Tap for more steps...
Step 6.5.2.1
Cancel the common factor of and .
Tap for more steps...
Step 6.5.2.1.1
Factor out of .
Step 6.5.2.1.2
Cancel the common factors.
Tap for more steps...
Step 6.5.2.1.2.1
Factor out of .
Step 6.5.2.1.2.2
Cancel the common factor.
Step 6.5.2.1.2.3
Rewrite the expression.
Step 6.5.2.1.2.4
Divide by .
Step 6.5.2.2
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 6.5.2.3
The exact value of is .
Step 6.5.2.4
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: None
Step 8