We think you are located in South Africa. Is this correct?

The Cosine Function

5.6 The cosine function (EMBH3)

Revision (EMBH4)

Functions of the form \(y = \cos \theta\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)

49522018b443b6844774536830b66724.png
  • The period is \(\text{360}\text{°}\) and the amplitude is \(\text{1}\).

  • Domain: \([\text{0}\text{°};\text{360}\text{°}]\)

    For \(y = \cos \theta\), the domain is \(\{ \theta: \theta \in \mathbb{R} \}\), however in this case, the domain has been restricted to the interval \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).

  • Range: \(\left[-1;1\right]\)

  • \(x\)-intercepts: \(\left(\text{90}\text{°};0\right)\), \(\left(\text{270}\text{°};0\right)\)

  • \(y\)-intercept: \(\left(\text{0}\text{°};1\right)\)

  • Maximum turning points: \(\left(\text{0}\text{°};1\right)\), \(\left(\text{360}\text{°};1\right)\)

  • Minimum turning point: \(\left(\text{180}\text{°};-1\right)\)

Functions of the form \(y = a \cos \theta + q\)

Cosine functions of the general form \(y = a \cos \theta + q\), where \(a\) and \(q\) are constants.

The effects of \(a\) and \(q\) on \(f(\theta) = a \cos \theta + q\):

  • The effect of \(q\) on vertical shift

    • For \(q>0\), \(f(\theta)\) is shifted vertically upwards by \(q\) units.

    • For \(q<0\), \(f(\theta)\) is shifted vertically downwards by \(q\) units.

  • The effect of \(a\) on shape

    • For \(a>1\), the amplitude of \(f(\theta)\) increases.

    • For \(0<a<1\), the amplitude of \(f(\theta)\) decreases.

    • For \(a<0\), there is a reflection about the \(x\)-axis.

    • For \(-1 < a < 0\), there is a reflection about the \(x\)-axis and the amplitude decreases.

    • For \(a < -1\), there is a reflection about the \(x\)-axis and the amplitude increases.

6944daa81d40275782fa10fda48ecaa5.png66a99e84dfca14a07d15ab0047d6fdc7.png

Revision

Exercise 5.24

On separate axes, accurately draw each of the following functions for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\):

  • Use tables of values if necessary.
  • Use graph paper if available.

For each function in the previous problem determine the following:

  • Period
  • Amplitude
  • Domain and range
  • \(x\)- and \(y\)-intercepts
  • Maximum and minimum turning points

\(y_1 = \cos \theta\)

4f5be4c27124d09642a724c585708fb3.png

\(y_2 = - 3 \cos \theta\)

37c02e97fb8d8540c720da77633c963f.png

\(y_3 = \cos \theta + 2\)

323e8afd585e1248c70b567b02c341d7.png

\(y_4 = \frac{1}{2} \cos \theta - 1\)

2935b945493c505e111c156e9eb504dd.png

Functions of the form \(y=\cos (k\theta)\) (EMBH5)

We now consider cosine functions of the form \(y = \cos k\theta\) and the effects of parameter \(k\).

The effects of \(k\) on a cosine graph

  1. Complete the following table for \(y_1 = \cos \theta\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\):
    θ\(-\text{360}\)\(\text{°}\)\(-\text{300}\)\(\text{°}\)\(-\text{240}\)\(\text{°}\)\(-\text{180}\)\(\text{°}\)\(-\text{120}\)\(\text{°}\)\(-\text{60}\)\(\text{°}\)\(\text{0}\)\(\text{°}\)
    \(\cos \theta\)
    θ\(\text{60}\)\(\text{°}\)\(\text{120}\)\(\text{°}\)\(\text{180}\)\(\text{°}\)\(\text{240}\)\(\text{°}\)\(\text{300}\)\(\text{°}\)\(\text{360}\)\(\text{°}\)
    \(\cos \theta\)
  2. Use the table of values to plot the graph of \(y_1 = \cos \theta\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).

  3. On the same system of axes, plot the following graphs:

    1. \(y_2 = \cos (-\theta)\)
    2. \(y_3 = \cos 3\theta\)
    3. \(y_4 = \cos \frac{3\theta}{4}\)
  4. Use your sketches of the functions above to complete the following table:

    \(y_1\)\(y_2\)\(y_3\)\(y_4\)
    period
    amplitude
    domain
    range
    maximum turning points
    minimum turning points
    \(y\)-intercept(s)
    \(x\)-intercept(s)
    effect of \(k\)
  5. What do you notice about \(y_1 = \cos \theta\) and \(y_2 = \cos (-\theta)\)?

  6. Is \(\cos (-\theta) = -\cos \theta\) a true statement? Explain your answer.

  7. Can you deduce a formula for determining the period of \(y = \cos k\theta\)?

The effect of the parameter \(k\) on \(y = \cos k\theta\)

The value of \(k\) affects the period of the cosine function.

  • For \(k > 0\):

    For \(k > 1\), the period of the cosine function decreases.

    For \(0 < k < 1\), the period of the cosine function increases.

  • For \(k < 0\):

    For \(-1 < k < 0\), the period increases.

    For \(k < -1\), the period decreases.

Negative angles: \[\cos (-\theta) = \cos \theta\] Notice that for negative values of \(\theta\), the graph is not reflected about the \(x\)-axis.

Calculating the period:

To determine the period of \(y = \cos k\theta\) we use, \[\text{Period } = \frac{\text{360}\text{°}}{|k|}\] where \(|k|\) is the absolute value of \(k\).

\(0 < k < 1\)

\(-1 < k < 0\)

58356ab256b4b9fa0156a64409f04592.pngddd3310bebaf8519b910d5fea228456d.png

\(k > 1\)

\(k < -1\)

984b092fb28a2d1de95c0e3dff84ceac.pngd8f9c98e34b1613992c75002fd8a56be.png

Worked example 22: Cosine function

  1. Sketch the following functions on the same set of axes for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\).
    1. \(y_1 = \cos \theta\)
    2. \(y_2 = \cos \frac{\theta}{2}\)
  2. For each function determine the following:

    1. Period
    2. Amplitude
    3. Domain and range
    4. \(x\)- and \(y\)-intercepts
    5. Maximum and minimum turning points

Examine the equations of the form \(y = \cos k\theta\)

Notice that for \(y_2 = \cos \frac{\theta}{2}\), \(k < 1\) therefore the period of the graph increases.

Complete a table of values

θ\(-\text{180}\)\(\text{°}\)\(-\text{135}\)\(\text{°}\)\(-\text{90}\)\(\text{°}\)\(-\text{45}\)\(\text{°}\)\(\text{0}\)\(\text{°}\)\(\text{45}\)\(\text{°}\)\(\text{90}\)\(\text{°}\)\(\text{135}\)\(\text{°}\)\(\text{180}\)\(\text{°}\)
\(\cos \theta\)\(-\text{1}\)\(-\text{0,71}\)\(\text{0}\)\(\text{0,71}\)\(\text{1}\)\(\text{0,71}\)\(\text{0}\)\(-\text{0,71}\)\(-\text{1}\)
\(\cos \frac{\theta}{2}\)\(\text{0}\)\(\text{0,38}\)\(\text{0,71}\)\(\text{0,92}\)\(\text{1}\)\(\text{0,92}\)\(\text{0,71}\)\(\text{0,38}\)\(\text{0}\)

Sketch the cosine graphs

02547eb34be12eaa2b97e43d7a3a9d15.png

Complete the table

\(y_1 = \cos \theta\)\(y_2 = \cos \frac{\theta}{2}\)
period\(\text{360}\text{°}\)\(\text{720}\text{°}\)
amplitude\(\text{1}\)\(\text{1}\)
domain\([-\text{180}\text{°};\text{180}\text{°}]\)\([-\text{180}\text{°};\text{180}\text{°}]\)
range\([-1;1]\)\([0;1]\)
maximum turning points\((\text{0}\text{°};1)\)\((\text{0}\text{°};1)\)
minimum turning points\((-\text{180}\text{°};-1) \text{ and } (\text{180}\text{°};-1)\)none
\(y\)-intercept(s)\((\text{0}\text{°};1)\)\((\text{0}\text{°};1)\)
\(x\)-intercept(s)\((-\text{90}\text{°};0) \text{ and } (\text{90}\text{°};0)\)\((-\text{180}\text{°};0) \text{ and } (\text{180}\text{°};0)\)

Discovering the characteristics

For functions of the general form: \(f(\theta) = y =\cos k\theta\):

Domain and range

The domain is \(\{ \theta: \theta \in \mathbb{R} \}\) because there is no value for \(\theta\) for which \(f(\theta)\) is undefined.

The range is \(\{ f(\theta): -1 \leq f(\theta) \leq 1, f(\theta) \in \mathbb{R} \}\) or \([-1;1]\).

Intercepts

The \(x\)-intercepts are determined by letting \(f(\theta) = 0\) and solving for \(\theta\).

The \(y\)-intercept is calculated by letting \(\theta = \text{0}\text{°}\) and solving for \(f(\theta)\). \begin{align*} y &= \cos k\theta \\ &= \cos \text{0}\text{°} \\ &= 1 \end{align*} This gives the point \((\text{0}\text{°};1)\).

Cosine functions of the form \(y = \cos k\theta\)

Exercise 5.25

Sketch the following functions for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\). For each graph determine:

  • Period
  • Amplitude
  • Domain and range
  • \(x\)- and \(y\)-intercepts
  • Maximum and minimum turning points

\(f(\theta) =\cos 2\theta\)

7ecdf8f7abbbdb5b4369db604ff346f3.png

For \(f(\theta) =\cos 2 \theta\):

\begin{align*} \text{Period: } & \text{180}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{180}\text{°};\text{180}\text{°}] \\ \text{Range: } & [-1;1] \\ x\text{-intercepts: } & (-\text{135}\text{°};0); (-\text{45}\text{°};0); (\text{45}\text{°};0); (\text{135}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};1) \\ \text{Max. turning point: } & (-\text{180}\text{°};1); (\text{0}\text{°};1); (\text{180}\text{°};1) \\ \text{Min. turning point: } & (-\text{90}\text{°};-1); (\text{90}\text{°};-1) \end{align*}

\(g(\theta) =\cos \frac{\theta}{3}\)

4b2fc9d9fad62693d6774617883aeba8.png

For \(g(\theta) =\cos \frac{\theta}{3}\):

\begin{align*} \text{Period: } & \text{1 080}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{180}\text{°};\text{180}\text{°}] \\ \text{Range: } & [\frac{1}{2};1] \\ x\text{-intercepts: } & \text{ none } \\ \text{Max. turning point: } & (\text{0}\text{°};1) \\ \text{Min. turning point: } & \text{ none } \end{align*}

\(h(\theta) =\cos (-2\theta)\)

0f99dbc1de1a0ec403ccefb9c6b37ceb.png

For \(h(\theta) =\cos (-2\theta)\):

\begin{align*} \text{Period: } & \text{180}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{180}\text{°};\text{180}\text{°}] \\ \text{Range: } & [-1;1] \\ x\text{-intercepts: } & (-\text{135}\text{°};0); (-\text{45}\text{°};0); (\text{45}\text{°};0); (\text{135}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};1) \\ \text{Max. turning point: } & (-\text{180}\text{°};1); (\text{0}\text{°};1); (\text{180}\text{°};1) \\ \text{Min. turning point: } & (-\text{90}\text{°};-1); (\text{90}\text{°};-1) \end{align*}

\(k(\theta) =\cos \frac{3\theta}{4}\)

9e49f88b1cc0b7952ab74134bb94f684.png

For \(k(\theta) =\cos \frac{3\theta}{4}\):

\begin{align*} \text{Period: } & \text{480}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{180}\text{°};\text{180}\text{°}] \\ \text{Range: } & [-\frac{1}{\sqrt{2}};1] \\ x\text{-intercepts: } & (-\text{120}\text{°};0); (\text{120}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};1) \\ \text{Max. turning point: } & (\text{0}\text{°};1) \\ \text{Min. turning point: } & \text{ none } \end{align*}

For each graph of the form \(f(\theta) =\cos k\theta\), determine the value of \(k\):

63a733c637cd152cc0985a2e01ce737f.png
\begin{align*} \text{Period } &= \frac{\text{720}\text{°}}{3 \text{ complete waves }} \\ &= \text{240}\text{°} \\ \therefore \frac{\text{360}\text{°}}{k} &= \text{240}\text{°} \\ \therefore k &= \frac{\text{360}\text{°}}{\text{240}\text{°} } \\ &= \frac{3}{2} \end{align*}
6758c51e8ce32689e29a0eb67700725a.png
\begin{align*} \text{For } y &= \cos \theta \\ 0 &= \cos \text{90}\text{°}\\ \text{So for } A(\text{135}\text{°};0) \qquad \text{90}\text{°} &= k \times \text{135}\text{°}\\ \therefore k &= \frac{\text{90}\text{°}}{\text{135}\text{°} } \\ &= \frac{2}{3} \end{align*}

Functions of the form \(y=\cos\left(\theta +p\right)\) (EMBH6)

We now consider cosine functions of the form \(y = \cos(\theta + p)\) and the effects of parameter \(p\).

The effects of \(p\) on a cosine graph

  1. On the same system of axes, plot the following graphs for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\):

    1. \(y_1 = \cos \theta\)
    2. \(y_2 = \cos (\theta - \text{90}\text{°})\)
    3. \(y_3 = \cos (\theta - \text{60}\text{°})\)
    4. \(y_4 = \cos (\theta + \text{90}\text{°})\)
    5. \(y_5 = \cos (\theta + \text{180}\text{°})\)
  2. Use your sketches of the functions above to complete the following table:

    \(y_1\)\(y_2\)\(y_3\)\(y_4\)\(y_5\)
    period
    amplitude
    domain
    range
    maximum turning points
    minimum turning points
    \(y\)-intercept(s)
    \(x\)-intercept(s)
    effect of \(p\)

The effect of the parameter on \(y = \cos(\theta + p)\)

The effect of \(p\) on the cosine function is a horizontal shift (or phase shift); the entire graph slides to the left or to the right.

  • For \(p > 0\), the graph of the cosine function shifts to the left by \(p\) degrees.

  • For \(p < 0\), the graph of the cosine function shifts to the right by \(p\) degrees.

\(p>0\)

\(p<0\)

8857f814c2f01495fa829db1b40ee28e.png7cdab00fd0c2ee79b876edc2a1de4ae9.png

Worked example 23: Cosine function

  1. Sketch the following functions on the same set of axes for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).
    1. \(y_1 = \cos \theta\)
    2. \(y_2 = \cos (\theta + \text{30}\text{°})\)
  2. For each function determine the following:

    1. Period
    2. Amplitude
    3. Domain and range
    4. \(x\)- and \(y\)-intercepts
    5. Maximum and minimum turning points

Examine the equations of the form \(y = \cos (\theta + p)\)

Notice that for \(y_1 = \cos \theta\) we have \(p = 0\) (no phase shift) and for \(y_2 = \cos (\theta + \text{30}\text{°})\), \(p < 0\) therefore the graph shifts to the left by \(\text{30}\text{°}\).

Complete a table of values

θ\(-\text{360}\)\(\text{°}\)\(-\text{270}\)\(\text{°}\)\(-\text{180}\)\(\text{°}\)\(-\text{90}\)\(\text{°}\)\(\text{0}\)\(\text{°}\)\(\text{90}\)\(\text{°}\)\(\text{180}\)\(\text{°}\)\(\text{270}\)\(\text{°}\)\(\text{360}\)\(\text{°}\)
\(\cos \theta\)\(\text{1}\)\(\text{0}\)\(-\text{1}\)\(\text{0}\)\(\text{1}\)\(\text{0}\)\(-\text{1}\)\(\text{0}\)\(\text{1}\)
\(\cos(\theta + \text{30}\text{°})\)\(\text{0,87}\)\(-\text{0,5}\)\(-\text{0,87}\)\(\text{0,5}\)\(\text{0,87}\)\(-\text{0,5}\)\(-\text{0,87}\)\(\text{0,5}\)\(\text{0,87}\)

Sketch the cosine graphs

d6700a39d7dab2a65b098027b2f13bcf.png

Complete the table

\(y_1\)\(y_2\)
period\(\text{360}\text{°}\)\(\text{360}\text{°}\)
amplitude\(\text{1}\)\(\text{1}\)
domain\([-\text{360}\text{°};\text{360}\text{°}]\)\([-\text{360}\text{°};\text{360}\text{°}]\)
range\([-1;1]\)\([-1;1]\)
maximum turning points\((-\text{360}\text{°};1)\), \((\text{0}\text{°};1)\) and \((\text{360}\text{°};1)\)\((-\text{30}\text{°};1)\) and \((\text{330}\text{°};1)\)
minimum turning points\((-\text{180}\text{°};-1)\) and \((\text{180}\text{°};-1)\)\((-\text{210}\text{°};-1)\) and \((\text{150}\text{°};-1)\)
\(y\)-intercept(s)\((\text{0}\text{°};0)\)\((\text{0}\text{°};\text{0,87})\)
\(x\)-intercept(s)\((-\text{270}\text{°};0)\), \((-\text{90}\text{°};0)\), \((\text{90}\text{°};0)\) and \((\text{270}\text{°};0)\)\((-\text{300}\text{°};0)\), \((-\text{120}\text{°};0)\), \((\text{60}\text{°};0)\) and \((\text{240}\text{°};0)\)

Discovering the characteristics

For functions of the general form: \(f(\theta) = y =\cos (\theta + p)\):

Domain and range

The domain is \(\{ \theta: \theta \in \mathbb{R} \}\) because there is no value for \(\theta\) for which \(f(\theta)\) is undefined.

The range is \(\{ f(\theta): -1 \leq f(\theta) \leq 1, f(\theta) \in \mathbb{R} \}\).

Intercepts

The \(x\)-intercepts are determined by letting \(f(\theta) = 0\) and solving for \(\theta\).

The \(y\)-intercept is calculated by letting \(\theta = \text{0}\text{°}\) and solving for \(f(\theta)\). \begin{align*} y &= \cos (\theta + p) \\ &= \cos (\text{0}\text{°} + p) \\ &= \cos p \end{align*} This gives the point \((\text{0}\text{°};\cos p)\).

Cosine functions of the form \(y = \cos (\theta + p)\)

Exercise 5.26

Sketch the following functions for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).

For each function, determine the following:

  • Period
  • Amplitude
  • Domain and range
  • \(x\)- and \(y\)-intercepts
  • Maximum and minimum turning points

\(f(\theta) =\cos (\theta + \text{45}\text{°})\)

4668040ccc402a9f27606d5d334820dd.png

\(g(\theta) =\cos (\theta - \text{30}\text{°})\)

a79798f0b9bb9d4cc64610d00cd8c4f8.png

\(h(\theta) =\cos (\theta + \text{60}\text{°})\)

9dcc10bfa606fb4afc2f52d661f2c589.png

Sketching cosine graphs (EMBH7)

Worked example 24: Sketching a cosine graph

Sketch the graph of \(f(\theta) = \cos (\text{180}\text{°} - 3\theta)\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).

Examine the form of the equation

Write the equation in the form \(y = \cos k(\theta + p)\). \begin{align*} f(\theta) &= \cos (\text{180}\text{°} - 3\theta)\\ &= \cos (-3\theta + \text{180}\text{°}) \\ &= \cos \left( -3(\theta - \text{60}\text{°}) \right) \\ &= \cos 3(\theta - \text{60}\text{°}) \end{align*}

To draw a graph of the above function, the standard cosine graph, \(y = \cos \theta\), must be changed in the following ways:

  • decrease the period by a factor of \(\text{3}\)
  • shift to the right by \(\text{60}\text{°}\).

Complete a table of values

θ\(\text{0}\)\(\text{°}\)\(\text{45}\)\(\text{°}\)\(\text{90}\)\(\text{°}\)\(\text{135}\)\(\text{°}\)\(\text{180}\)\(\text{°}\)\(\text{225}\)\(\text{°}\)\(\text{270}\)\(\text{°}\)\(\text{315}\)\(\text{°}\)\(\text{360}\)\(\text{°}\)
\(f(\theta)\)\(-\text{1}\)\(\text{0,71}\)\(\text{0}\)\(-\text{0,71}\)\(\text{1}\)\(-\text{0,71}\)\(\text{0}\)\(\text{0,71}\)\(-\text{1}\)

Plot the points and join with a smooth curve

bc7cd42cb394fa543b55bb9acb9aa867.png

Period: \(\text{120}\)\(\text{°}\)

Amplitude: \(\text{1}\)

Domain: \([\text{0}\text{°};\text{360}\text{°}]\)

Range: \([-1;1]\)

Maximum turning point: \((\text{60}\text{°};1)\), \((\text{180}\text{°};1)\) and \((\text{300}\text{°};1)\)

Minimum turning point: \((\text{0}\text{°}; -1)\), \((\text{120}\text{°};-1)\), \((\text{240}\text{°};-1)\) and \((\text{360}\text{°};-1)\)

\(y\)-intercepts: \((\text{0}\text{°};-1)\)

\(x\)-intercept: \((\text{30}\text{°};0)\), \((\text{90}\text{°};0)\), \((\text{150}\text{°};0)\), \((\text{210}\text{°};0)\), \((\text{270}\text{°};0)\) and \((\text{330}\text{°};0)\)

Worked example 25: Finding the equation of a cosine graph

Given the graph of \(y = a \cos (k\theta + p)\), determine the values of \(a\), \(k\), \(p\) and the minimum turning point.

c7f1dc7aa6978ba9fe531bcb9e924c50.png

Determine the value of \(k\)

From the sketch we see that the period of the graph is \(\text{360}\text{°}\), therefore \(k = 1\).

\[y = a \cos ( \theta + p)\]

Determine the value of \(a\)

From the sketch we see that the maximum turning point is \((\text{45}\text{°};2)\), so we know that the amplitude of the graph is \(\text{2}\) and therefore \(a = 2\).

\[y = 2 \cos ( \theta + p)\]

Determine the value of \(p\)

Compare the given graph with the standard cosine function \(y = \cos \theta\) and notice the difference in the maximum turning points. We see that the given function has been shifted to the right by \(\text{45}\)\(\text{°}\), therefore \(p = \text{45}\text{°}\).

\[y = 2 \cos ( \theta - \text{45}\text{°})\]

Determine the minimum turning point

At the minimum turning point, \(y = -2\):

\begin{align*} y &= 2 \cos ( \theta - \text{45}\text{°}) \\ -2 &= 2 \cos ( \theta - \text{45}\text{°}) \\ -1 &= \cos ( \theta - \text{45}\text{°}) \\ \cos^{-1}(-1) &= \theta - \text{45}\text{°} \\ \text{180}\text{°} &= \theta - \text{45}\text{°} \\ \text{225}\text{°} &= \theta \end{align*}

This gives the point \((\text{225}\text{°};-2)\).

The cosine function

Exercise 5.27

Sketch the following graphs on separate axes:

\(y = \cos (\theta + \text{15}\text{°})\) for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\)

c0985f83282677b24577678613549a50.png

\(f(\theta) = \frac{1}{3} \cos (\theta - \text{60}\text{°})\) for \(-\text{90}\text{°} \leq \theta \leq \text{90}\text{°}\)

1b00154d916ba560d4a4c8f099f5bf0c.png

\(y = -2 \cos \theta\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)

558877ea3d33d60228182a1db71af0e5.png

\(y = \cos (\text{30}\text{°} - \theta)\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\)

\begin{align*} y &= \cos (\text{30}\text{°} - \theta) \\ &= \cos \left( -(\theta - \text{30}\text{°}) \right) \\ &= \cos \left(\theta - \text{30}\text{°} \right) \end{align*} 820155870903b2ce6a3c6f5d561cf131.png

\(g(\theta) = 1 + \cos (\theta - \text{90}\text{°})\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)

f404be7db91a953f33288ca035198cc6.png

\(y = \cos (2 \theta + \text{60}\text{°})\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\)

def0cddd8564099cf7fee09d485a0000.png

Two girls are given the following graph:

b9a678ad34313e227a39e31bb17959fd.png

Audrey decides that the equation for the graph is a cosine function of the form \(y = a \cos \theta\). Determine the value of \(a\).

\(a = -1\)

Megan thinks that the equation for the graph is a cosine function of the form \(y = \cos (\theta + p)\). Determine the value of \(p\).

\(p = -\text{180}\text{°}\)

What can they conclude?

\(\cos (\theta - \text{180}\text{°}) = -\cos \theta\)