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# Trigonometry Examples

Use the form to find the variables used to find the amplitude, period, phase shift, and vertical shift.

Find the amplitude .

Amplitude:

The period of the function can be calculated using .

Period:

Replace with in the formula for period.

Period:

Solve the equation.

The absolute value is the distance between a number and zero. The distance between and is .

Period:

Cancel the common factor of .

Cancel the common factor.

Period:

Divide by .

Period:

Period:

Period:

Period:

The phase shift of the function can be calculated from .

Phase Shift:

Replace the values of and in the equation for phase shift.

Phase Shift:

Divide by .

Phase Shift:

Phase Shift:

Find the vertical shift .

Vertical Shift:

List the properties of the trigonometric function.

Amplitude:

Period:

Phase Shift: ( to the right)

Vertical Shift:

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Multiply by .

The exact value of is .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

The exact value of is .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

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.

The exact value of is .

Multiply by .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

The exact value of is .

The final answer is .

Find the point at .

Replace the variable with in the expression.

Simplify the result.

is a full rotation so replace with .

The exact value of is .

The final answer is .

List the points in a table.

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: