Trigonometry Examples

Graph h(x)=0.5sin(0.25x+0.75pi)-5
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.1.4
Replace with an approximation.
Step 3.1.5
Multiply by .
Step 3.1.6
Divide 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
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.4
Replace with an approximation.
Step 3.2.5
Multiply by .
Step 3.2.6
Divide 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
Divide by .
Phase Shift:
Phase Shift:
Step 5
List the properties of the trigonometric function.
Amplitude:
Period:
Phase Shift: ( to the left)
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
Multiply by .
Step 6.1.2.1.2
Add and .
Step 6.1.2.1.3
The exact value of is .
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Step 6.1.2.1.3.1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 6.1.2.1.3.2
Apply the sine half-angle identity.
Step 6.1.2.1.3.3
Change the to because sine is positive in the first quadrant.
Step 6.1.2.1.3.4
Simplify .
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Step 6.1.2.1.3.4.1
The exact value of is .
Step 6.1.2.1.3.4.2
Multiply by .
Step 6.1.2.1.3.4.3
Subtract from .
Step 6.1.2.1.3.4.4
Divide by .
Step 6.1.2.1.3.4.5
Rewrite as .
Step 6.1.2.1.3.4.6
Pull terms out from under the radical, assuming positive real numbers.
Step 6.1.2.1.4
Multiply by .
Step 6.1.2.2
Subtract from .
Step 6.1.2.3
The final answer is .
Step 6.2
Find the point at .
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Step 6.2.1
Replace the variable with in the expression.
Step 6.2.2
Simplify the result.
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Step 6.2.2.1
Simplify each term.
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Step 6.2.2.1.1
Multiply by .
Step 6.2.2.1.2
Add and .
Step 6.2.2.1.3
Multiply by .
Step 6.2.2.2
Subtract from .
Step 6.2.2.3
The final answer is .
Step 6.3
Find the point at .
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Step 6.3.1
Replace the variable with in the expression.
Step 6.3.2
Simplify the result.
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Step 6.3.2.1
Simplify each term.
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Step 6.3.2.1.1
Multiply by .
Step 6.3.2.1.2
Add and .
Step 6.3.2.1.3
Multiply by .
Step 6.3.2.2
Subtract from .
Step 6.3.2.3
The final answer is .
Step 6.4
Find the point at .
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Step 6.4.1
Replace the variable with in the expression.
Step 6.4.2
Simplify the result.
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Step 6.4.2.1
Simplify each term.
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Step 6.4.2.1.1
Multiply by .
Step 6.4.2.1.2
Add and .
Step 6.4.2.1.3
Multiply by .
Step 6.4.2.2
Subtract from .
Step 6.4.2.3
The final answer is .
Step 6.5
Find the point at .
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Step 6.5.1
Replace the variable with in the expression.
Step 6.5.2
Simplify the result.
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Step 6.5.2.1
Simplify each term.
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Step 6.5.2.1.1
Multiply by .
Step 6.5.2.1.2
Add and .
Step 6.5.2.1.3
Multiply by .
Step 6.5.2.2
Subtract from .
Step 6.5.2.3
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 left)
Vertical Shift:
Step 8