Trigonometry Examples

Graph h(t)=40sin((5pi)/4t-pi/2)+55
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 using the formula .
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Step 3.1
Find the period of .
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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 .
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Step 3.1.5.1
Factor out of .
Step 3.1.5.2
Factor out of .
Step 3.1.5.3
Cancel the common factor.
Step 3.1.5.4
Rewrite the expression.
Step 3.1.6
Combine and .
Step 3.1.7
Multiply by .
Step 3.2
Find the period of .
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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 .
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Step 3.2.5.1
Factor out of .
Step 3.2.5.2
Factor out of .
Step 3.2.5.3
Cancel the common factor.
Step 3.2.5.4
Rewrite the expression.
Step 3.2.6
Combine and .
Step 3.2.7
Multiply by .
Step 3.3
The period of addition/subtraction of trig functions is the maximum of the individual periods.
Step 4
Find the phase shift using the formula .
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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
Cancel the common factor of .
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Step 4.4.1
Factor out of .
Phase Shift:
Step 4.4.2
Cancel the common factor.
Phase Shift:
Step 4.4.3
Rewrite the expression.
Phase Shift:
Phase Shift:
Step 4.5
Cancel the common factor of .
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Step 4.5.1
Factor out of .
Phase Shift:
Step 4.5.2
Cancel the common factor.
Phase Shift:
Step 4.5.3
Rewrite the expression.
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
Select a few points to graph.
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Step 6.1
Find the point at .
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Step 6.1.1
Replace the variable with in the expression.
Step 6.1.2
Simplify the result.
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Step 6.1.2.1
Simplify each term.
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Step 6.1.2.1.1
Simplify each term.
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Step 6.1.2.1.1.1
Simplify the numerator.
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Step 6.1.2.1.1.1.1
Combine and .
Step 6.1.2.1.1.1.2
Combine and .
Step 6.1.2.1.1.2
Multiply by .
Step 6.1.2.1.1.3
Reduce the expression by cancelling the common factors.
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Step 6.1.2.1.1.3.1
Reduce the expression by cancelling the common factors.
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Step 6.1.2.1.1.3.1.1
Factor out of .
Step 6.1.2.1.1.3.1.2
Factor out of .
Step 6.1.2.1.1.3.1.3
Cancel the common factor.
Step 6.1.2.1.1.3.1.4
Rewrite the expression.
Step 6.1.2.1.1.3.2
Divide by .
Step 6.1.2.1.1.4
Cancel the common factor of and .
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Step 6.1.2.1.1.4.1
Factor out of .
Step 6.1.2.1.1.4.2
Cancel the common factors.
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Step 6.1.2.1.1.4.2.1
Factor out of .
Step 6.1.2.1.1.4.2.2
Cancel the common factor.
Step 6.1.2.1.1.4.2.3
Rewrite the expression.
Step 6.1.2.1.1.4.2.4
Divide by .
Step 6.1.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 6.1.2.1.3
Combine and .
Step 6.1.2.1.4
Combine the numerators over the common denominator.
Step 6.1.2.1.5
Simplify the numerator.
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Step 6.1.2.1.5.1
Multiply by .
Step 6.1.2.1.5.2
Subtract from .
Step 6.1.2.1.6
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.
Step 6.1.2.1.7
The exact value of is .
Step 6.1.2.1.8
Multiply .
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Step 6.1.2.1.8.1
Multiply by .
Step 6.1.2.1.8.2
Multiply by .
Step 6.1.2.2
Add and .
Step 6.1.2.3
The final answer is .
Step 6.2
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