Trigonometry Examples

$f (t) = 40 \cos (80 t + 20)$

Step 1

Use the form $a \cos (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 40$

$b = 80$

$c = - 20$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $40$

Step 3

Find the period of $40 \cos (80 x + 20)$ .

Tap for more steps...

Step 3.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $80$ in the formula for period.

$\frac{2 π}{| 80 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $0$ and $80$ is $80$ .

$\frac{2 π}{80}$

Step 3.4

Cancel the common factor of $2$ and $80$ .

Tap for more steps...

Step 3.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{80}$

Step 3.4.2

Cancel the common factors.

Tap for more steps...

Step 3.4.2.1

Factor $2$ out of $80$ .

$\frac{2 π}{2 \cdot 40}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 40}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{40}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

Tap for more steps...

Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{- 20}{80}$

Step 4.3

Cancel the common factor of $- 20$ and $80$ .

Tap for more steps...

Step 4.3.1

Factor $20$ out of $- 20$ .

Phase Shift: $\frac{20 (- 1)}{80}$

Step 4.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.1

Factor $20$ out of $80$ .

Phase Shift: $\frac{20 \cdot - 1}{20 \cdot 4}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift: $\frac{20 \cdot - 1}{20 \cdot 4}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift: $\frac{- 1}{4}$

Step 4.4

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: $40$

Period: $\frac{π}{40}$

Phase Shift: $- \frac{1}{4}$ ( $\frac{1}{4}$ to the left)

Vertical Shift: None

Step 6

Select a few points to graph.

Tap for more steps...

Step 6.1

Find the point at $x = - \frac{1}{4}$ .

Tap for more steps...

Step 6.1.1

Replace the variable $x$ with $- \frac{1}{4}$ in the expression.

$f (- \frac{1}{4}) = 40 \cos (80 (- \frac{1}{4}) + 20)$

Step 6.1.2

Simplify the result.

Tap for more steps...

Step 6.1.2.1

Simplify each term.

Tap for more steps...

Step 6.1.2.1.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.1.2.1.1.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (- \frac{1}{4}) = 40 \cos (80 (\frac{- 1}{4}) + 20)$

Step 6.1.2.1.1.2

Factor $4$ out of $80$ .

$f (- \frac{1}{4}) = 40 \cos (4 (20) (\frac{- 1}{4}) + 20)$

Step 6.1.2.1.1.3

Cancel the common factor.

$f (- \frac{1}{4}) = 40 \cos (4 \cdot (20 (\frac{- 1}{4})) + 20)$

Step 6.1.2.1.1.4

Rewrite the expression.

$f (- \frac{1}{4}) = 40 \cos (20 \cdot - 1 + 20)$

Step 6.1.2.1.2

Multiply $20$ by $- 1$ .

$f (- \frac{1}{4}) = 40 \cos (- 20 + 20)$

Step 6.1.2.2

Add $- 20$ and $20$ .

$f (- \frac{1}{4}) = 40 \cos (0)$

Step 6.1.2.3

The exact value of $\cos (0)$ is $1$ .

$f (- \frac{1}{4}) = 40 \cdot 1$

Step 6.1.2.4

Multiply $40$ by $1$ .

$f (- \frac{1}{4}) = 40$

Step 6.1.2.5

The final answer is $40$ .

$40$

Step 6.2

Find the point at $x = \frac{π}{160} - \frac{1}{4}$ .

Tap for more steps...

Step 6.2.1

Replace the variable $x$ with $\frac{π}{160} - \frac{1}{4}$ in the expression.

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{160} - \frac{1}{4}) + 20)$

Step 6.2.2

Simplify the result.

Tap for more steps...

Step 6.2.2.1

Simplify each term.

Tap for more steps...

Step 6.2.2.1.1

Apply the distributive property.

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{160}) + 80 (- \frac{1}{4}) + 20)$

Step 6.2.2.1.2

Cancel the common factor of $80$ .

Tap for more steps...

Step 6.2.2.1.2.1

Factor $80$ out of $160$ .

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{80 (2)}) + 80 (- \frac{1}{4}) + 20)$

Step 6.2.2.1.2.2

Cancel the common factor.

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{80 \cdot 2}) + 80 (- \frac{1}{4}) + 20)$

Step 6.2.2.1.2.3

Rewrite the expression.

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2} + 80 (- \frac{1}{4}) + 20)$

Step 6.2.2.1.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.2.1.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2} + 80 (\frac{- 1}{4}) + 20)$

Step 6.2.2.1.3.2

Factor $4$ out of $80$ .

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2} + 4 (20) (\frac{- 1}{4}) + 20)$

Step 6.2.2.1.3.3

Cancel the common factor.

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2} + 4 \cdot (20 (\frac{- 1}{4})) + 20)$

Step 6.2.2.1.3.4

Rewrite the expression.

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2} + 20 \cdot - 1 + 20)$

Step 6.2.2.1.4

Multiply $20$ by $- 1$ .

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2} - 20 + 20)$

Step 6.2.2.2

Simplify by adding numbers.

Tap for more steps...

Step 6.2.2.2.1

Add $- 20$ and $20$ .

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2} + 0)$

Step 6.2.2.2.2

Add $\frac{π}{2}$ and $0$ .

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2})$

Step 6.2.2.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{π}{160} - \frac{1}{4}) = 40 \cdot 0$

Step 6.2.2.4

Multiply $40$ by $0$ .

$f (\frac{π}{160} - \frac{1}{4}) = 0$

Step 6.2.2.5

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{π}{80} - \frac{1}{4}$ .

Tap for more steps...

Step 6.3.1

Replace the variable $x$ with $\frac{π}{80} - \frac{1}{4}$ in the expression.

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{80} - \frac{1}{4}) + 20)$

Step 6.3.2

Simplify the result.

Tap for more steps...

Step 6.3.2.1

Simplify each term.

Tap for more steps...

Step 6.3.2.1.1

Apply the distributive property.

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{80}) + 80 (- \frac{1}{4}) + 20)$

Step 6.3.2.1.2

Cancel the common factor of $80$ .

Tap for more steps...

Step 6.3.2.1.2.1

Cancel the common factor.

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{80}) + 80 (- \frac{1}{4}) + 20)$

Step 6.3.2.1.2.2

Rewrite the expression.

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π + 80 (- \frac{1}{4}) + 20)$

Step 6.3.2.1.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.3.2.1.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π + 80 (\frac{- 1}{4}) + 20)$

Step 6.3.2.1.3.2

Factor $4$ out of $80$ .

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π + 4 (20) (\frac{- 1}{4}) + 20)$

Step 6.3.2.1.3.3

Cancel the common factor.

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π + 4 \cdot (20 (\frac{- 1}{4})) + 20)$

Step 6.3.2.1.3.4

Rewrite the expression.

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π + 20 \cdot - 1 + 20)$

Step 6.3.2.1.4

Multiply $20$ by $- 1$ .

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π - 20 + 20)$

Step 6.3.2.2

Simplify by adding numbers.

Tap for more steps...

Step 6.3.2.2.1

Add $- 20$ and $20$ .

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π + 0)$

Step 6.3.2.2.2

Add $π$ and $0$ .

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cos (π)$

Step 6.3.2.3

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.

$f (\frac{π}{80} - \frac{1}{4}) = 40 (- \cos (0))$

Step 6.3.2.4

The exact value of $\cos (0)$ is $1$ .

$f (\frac{π}{80} - \frac{1}{4}) = 40 (- 1 \cdot 1)$

Step 6.3.2.5

Multiply $40 (- 1 \cdot 1)$ .

Tap for more steps...

Step 6.3.2.5.1

Multiply $- 1$ by $1$ .

$f (\frac{π}{80} - \frac{1}{4}) = 40 \cdot - 1$

Step 6.3.2.5.2

Multiply $40$ by $- 1$ .

$f (\frac{π}{80} - \frac{1}{4}) = - 40$

Step 6.3.2.6

The final answer is $- 40$ .

$- 40$

Step 6.4

Find the point at $x = \frac{3 π}{160} - \frac{1}{4}$ .

Tap for more steps...

Step 6.4.1

Replace the variable $x$ with $\frac{3 π}{160} - \frac{1}{4}$ in the expression.

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{3 π}{160} - \frac{1}{4}) + 20)$

Step 6.4.2

Simplify the result.

Tap for more steps...

Step 6.4.2.1

Simplify each term.

Tap for more steps...

Step 6.4.2.1.1

Apply the distributive property.

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{3 π}{160}) + 80 (- \frac{1}{4}) + 20)$

Step 6.4.2.1.2

Cancel the common factor of $80$ .

Tap for more steps...

Step 6.4.2.1.2.1

Factor $80$ out of $160$ .

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{3 π}{80 (2)}) + 80 (- \frac{1}{4}) + 20)$

Step 6.4.2.1.2.2

Cancel the common factor.

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (80 (\frac{3 π}{80 \cdot 2}) + 80 (- \frac{1}{4}) + 20)$

Step 6.4.2.1.2.3

Rewrite the expression.

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2} + 80 (- \frac{1}{4}) + 20)$

Step 6.4.2.1.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.4.2.1.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2} + 80 (\frac{- 1}{4}) + 20)$

Step 6.4.2.1.3.2

Factor $4$ out of $80$ .

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2} + 4 (20) (\frac{- 1}{4}) + 20)$

Step 6.4.2.1.3.3

Cancel the common factor.

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2} + 4 \cdot (20 (\frac{- 1}{4})) + 20)$

Step 6.4.2.1.3.4

Rewrite the expression.

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2} + 20 \cdot - 1 + 20)$

Step 6.4.2.1.4

Multiply $20$ by $- 1$ .

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2} - 20 + 20)$

Step 6.4.2.2

Simplify by adding numbers.

Tap for more steps...

Step 6.4.2.2.1

Add $- 20$ and $20$ .

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2} + 0)$

Step 6.4.2.2.2

Add $\frac{3 π}{2}$ and $0$ .

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{3 π}{2})$

Step 6.4.2.3

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

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cos (\frac{π}{2})$

Step 6.4.2.4

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{3 π}{160} - \frac{1}{4}) = 40 \cdot 0$

Step 6.4.2.5

Multiply $40$ by $0$ .

$f (\frac{3 π}{160} - \frac{1}{4}) = 0$

Step 6.4.2.6

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = \frac{π}{40} - \frac{1}{4}$ .

Tap for more steps...

Step 6.5.1

Replace the variable $x$ with $\frac{π}{40} - \frac{1}{4}$ in the expression.

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{40} - \frac{1}{4}) + 20)$

Step 6.5.2

Simplify the result.

Tap for more steps...

Step 6.5.2.1

Simplify each term.

Tap for more steps...

Step 6.5.2.1.1

Apply the distributive property.

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (80 (\frac{π}{40}) + 80 (- \frac{1}{4}) + 20)$

Step 6.5.2.1.2

Cancel the common factor of $40$ .

Tap for more steps...

Step 6.5.2.1.2.1

Factor $40$ out of $80$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (40 (2) (\frac{π}{40}) + 80 (- \frac{1}{4}) + 20)$

Step 6.5.2.1.2.2

Cancel the common factor.

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (40 \cdot (2 (\frac{π}{40})) + 80 (- \frac{1}{4}) + 20)$

Step 6.5.2.1.2.3

Rewrite the expression.

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π + 80 (- \frac{1}{4}) + 20)$

Step 6.5.2.1.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.5.2.1.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π + 80 (\frac{- 1}{4}) + 20)$

Step 6.5.2.1.3.2

Factor $4$ out of $80$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π + 4 (20) (\frac{- 1}{4}) + 20)$

Step 6.5.2.1.3.3

Cancel the common factor.

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π + 4 \cdot (20 (\frac{- 1}{4})) + 20)$

Step 6.5.2.1.3.4

Rewrite the expression.

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π + 20 \cdot - 1 + 20)$

Step 6.5.2.1.4

Multiply $20$ by $- 1$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π - 20 + 20)$

Step 6.5.2.2

Simplify by adding numbers.

Tap for more steps...

Step 6.5.2.2.1

Add $- 20$ and $20$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π + 0)$

Step 6.5.2.2.2

Add $2 π$ and $0$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (2 π)$

Step 6.5.2.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cos (0)$

Step 6.5.2.4

The exact value of $\cos (0)$ is $1$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40 \cdot 1$

Step 6.5.2.5

Multiply $40$ by $1$ .

$f (\frac{π}{40} - \frac{1}{4}) = 40$

Step 6.5.2.6

The final answer is $40$ .

$40$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{1}{4} & 40 \\ \frac{π}{160} - \frac{1}{4} & 0 \\ \frac{π}{80} - \frac{1}{4} & - 40 \\ \frac{3 π}{160} - \frac{1}{4} & 0 \\ \frac{π}{40} - \frac{1}{4} & 40 \end{array}$

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: $40$

Period: $\frac{π}{40}$

Phase Shift: $- \frac{1}{4}$ ( $\frac{1}{4}$ to the left)

Vertical Shift: None

Step 8