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5.5 The sine function

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5.4 Exponential functions
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5.6 The cosine function

5.5 The sine function (EMBGW)

Revision (EMBGX)

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

a94c55f8b736b9eb466eb6a71a58fc08.png

  • Period of one complete wave is \(\text{360}\)\(\text{°}\).

  • Amplitude is the maximum height of the wave above and below the \(x\)-axis and is always positive. Amplitude = \(\text{1}\).

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

    For \(y = \sin \theta\), 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: \(\left(\text{0}\text{°};0\right)\), \(\left(\text{180}\text{°};0\right)\), \(\left(\text{360}\text{°};0\right)\)

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

  • Maximum turning point: \(\left(\text{90}\text{°};1\right)\)

  • Minimum turning point: \(\left(\text{270}\text{°};-1\right)\)

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

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

  • 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.

832943a9daa624985ea283bc17078de2.pngbd862dc1f362fb9b44c3c4bbfbfeb431.png
temp text

Revision

Textbook Exercise 5.20

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 also determine the following:

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

\(y_1 = \sin \theta\)

0a62cd55a11424d9c67ad053aeb91db4.png

\(y_2 = - 2 \sin \theta\)

ad8404fa98ea301bdd1d5a92bcfdb519.png

\(y_3 = \sin \theta + 1\)

70a959df0f93f2a9171207c7fb3c1b70.png

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

22e20f86c50fed5a6fb1787328ba5dc5.png

Functions of the form \(y = \sin k\theta\) (EMBGY)

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

  1. Complete the following table for \(y_1 = \sin \theta\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\):
    \(θ\) \(-\text{360}\)\(\text{°}\) \(-\text{270}\)\(\text{°}\) \(-\text{180}\)\(\text{°}\) \(-\text{90}\)\(\text{°}\) \(\text{0}\)\(\text{°}\) \(\text{90}\)\(\text{°}\) \(\text{180}\)\(\text{°}\) \(\text{270}\)\(\text{°}\) \(\text{360}\)\(\text{°}\)
    \(\sin \theta\)

  2. Use the table of values to plot the graph of \(y_1 = \sin \theta\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).

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

    1. \(y_2 = \sin (-\theta)\)
    2. \(y_3 = \sin 2\theta\)
    3. \(y_4 = \sin \frac{\theta}{2}\)

  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 \(y_1 = \sin \theta\) and \(y_2 = \sin (-\theta)\)?

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

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

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

The value of \(k\) affects the period of the sine function. If \(k\) is negative, then the graph is reflected about the \(y\)-axis.

  • For \(k > 0\):

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

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

  • For \(k < 0\):

    For \(-1 < k < 0\), the graph is reflected about the \(y\)-axis and the period increases.

    For \(k < -1\), the graph is reflected about the \(y\)-axis and the period decreases.

Negative angles: \[\sin (-\theta) = -\sin \theta\]

Calculating the period:

To determine the period of \(y = \sin k\theta\) we use, \[\text{Period } = \frac{\text{360}\text{°}}{|k|}\] where \(|k|\) is the absolute value of \(k\) (this means that \(k\) is always considered to be positive).

\(0 < k < 1\)

\(-1 < k < 0\)
1b39c5955de4ff4769778a00dc6f9bec.png 9489eeaa93dfc8a3ea08de3bbb12e149.png

\(k > 1\)

\(k < -1\)
aed1a16b6d01b9250823e94de6445009.png 4c3ec29fdca6d6878e134da51a3c64d7.png

Worked example 18: Sine function

  1. Sketch the following functions on the same set of axes for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\).
    1. \(y_1 = \sin \theta\)
    2. \(y_2 = \sin \frac{3\theta}{2}\)

  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 \(y = \sin k\theta\)

Notice that \(k > 1\) for \(y_2 = \sin \frac{3\theta}{2}\), therefore the period of the graph decreases.

Complete a table of values

\(θ\) \(-\text{180}\)\(\text{°}\) \(-\text{135}\)\(\text{°}\) \(-\text{90}\)\(\text{°}\) \(-\text{45}\)\(\text{°}\) \(\text{0}\)\(\text{°}\) \(\text{45}\)\(\text{°}\) \(\text{90}\)\(\text{°}\) \(\text{135}\)\(\text{°}\) \(\text{180}\)\(\text{°}\)
\(\sin \theta\) \(\text{0}\) \(-\text{0,71}\) \(-\text{1}\) \(-\text{0,71}\) \(\text{0}\) \(\text{0,71}\) \(\text{1}\) \(\text{0,71}\) \(\text{0}\)
\(\sin \frac{3\theta}{2}\) \(\text{1}\) \(\text{0,38}\) \(-\text{0,71}\) \(-\text{0,92}\) \(\text{0}\) \(\text{0,92}\) \(\text{0,71}\) \(-\text{0,38}\) \(-\text{1}\)

Sketch the sine graphs

11d6bcc079858c9218c92936fe08f725.png

Complete the table

\(y_1 = \sin \theta\) \(y_2 = \sin \frac{3\theta}{2}\)
period \(\text{360}\)\(\text{°}\) \(\text{240}\)\(\text{°}\)
amplitude \(\text{1}\) \(\text{1}\)
domain \([-\text{180}\text{°};\text{180}\text{°}]\) \([-\text{180}\text{°};\text{180}\text{°}]\)
range \([-1;1]\) \([-1;1]\)
maximum turning points \((\text{90}\text{°};1)\) \((-\text{180}\text{°};1)\) and \((\text{60}\text{°};1)\)
minimum turning points \((-\text{90}\text{°};-1)\) \((-\text{60}\text{°};-1) \text{ and } (\text{180}\text{°};1)\)
\(y\)-intercept(s) \((\text{0}\text{°};0)\) \((\text{0}\text{°};0)\)
\(x\)-intercept(s) \((-\text{180}\text{°};0)\), \((\text{0}\text{°};0)\) and \((\text{180}\text{°};0)\) \((-\text{120}\text{°};0)\), \((\text{0}\text{°};0)\) and \((\text{120}\text{°};0)\)

Discovering the characteristics

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

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 &= \sin k\theta \\ &= \sin \text{0}\text{°} \\ &= 0 \end{align*} This gives the point \((\text{0}\text{°};0)\).

temp text

Sine functions of the form \(y = \sin k\theta\)

Textbook Exercise 5.21

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

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

\(f(\theta) =\sin 3\theta\)

e174c033ca61506fb7a2c3fe8bf14358.png

For \(f(\theta) =\sin 3\theta\):

\begin{align*} \text{Period: } & \text{120}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{180}\text{°};\text{180}\text{°}] \\ \text{Range: } & [-1;1] \\ x\text{-intercepts: } & (-\text{180}\text{°};0); (-\text{120}\text{°};0); (-\text{60}\text{°};0); \\ & (\text{0}\text{°};0); (\text{60}\text{°};0); (\text{120}\text{°};0); (\text{180}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};0) \\ \text{Max. turning point: } & (-\text{90}\text{°};1); (\text{30}\text{°};1); (\text{150}\text{°};1) \\ \text{Min. turning point: } & (-\text{150}\text{°};-1); (-\text{30}\text{°};-1); (\text{90}\text{°};-1) \end{align*}

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

573392583889890bfa94b3628b3829f4.png

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

\begin{align*} \text{Period: } & \text{1 080}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{180}\text{°};\text{180}\text{°}] \\ \text{Range: } & [-\text{0,87};\text{0,87}] \\ x\text{-intercepts: } & \text{ none } \\ y\text{-intercepts: } & (\text{0}\text{°};0) \\ \text{Max. turning point: } & \text{ none } \\ \text{Min. turning point: } & \text{ none } \end{align*}

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

b74231008486c57e3585ce307421a87e.png

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

\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{180}\text{°};0); (-\text{90}\text{°};0); (\text{0}\text{°};0); (\text{90}\text{°};0); (\text{180}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};0) \\ \text{Max. turning point: } & (-\text{45}\text{°};1); (\text{135}\text{°};1) \\ \text{Min. turning point: } & (-\text{135}\text{°};-1); (\text{45}\text{°};-1); \end{align*}

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

8894e5535d74972d3a41a70c1a0c38c3.png

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

\begin{align*} \text{Period: } & \text{480}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{180}\text{°};\text{180}\text{°}] \\ \text{Range: } & [-1;1] \\ x\text{-intercepts: } & (\text{0}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};0) \\ \text{Max. turning point: } & (\text{120}\text{°};1) \\ \text{Min. turning point: } & (-\text{120}\text{°};-1); \end{align*}

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

910b6612a9dc0c066c77a9929b689b3a.png
\begin{align*} \text{Period } &= \text{180}\text{°} \\ \therefore \frac{\text{360}\text{°}}{k} &= \text{180}\text{°} \\ k &= \frac{\text{360}\text{°}}{\text{180}\text{°}} \\ \therefore k &= 2 \end{align*}
e78bca94e51531e8475e28eab21de528.png
\begin{align*} \text{Period } &= \text{270}\text{°} \\ \therefore \frac{\text{360}\text{°}}{k} &= \text{270}\text{°} \\ k &= \frac{\text{360}\text{°}}{\text{270}\text{°}} \\ \therefore k &= \frac{3}{4} \\ \text{and graph is reflected about the } x-\text{axis } \therefore k &= -\frac{3}{4} \end{align*}

Functions of the form \(y = \sin(\theta + p)\) (EMBGZ)

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

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

    1. \(y_1 = \sin \theta\)
    2. \(y_2 = \sin (\theta - \text{90}\text{°})\)
    3. \(y_3 = \sin (\theta - \text{60}\text{°})\)
    4. \(y_4 = \sin (\theta + \text{90}\text{°})\)
    5. \(y_5 = \sin (\theta + \text{180}\text{°})\)

  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 \(y = \sin(\theta + p)\)

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

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

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

\(p>0\)

\(p<0\)
b5cfcf9c5317800a74560ae08b597aad.png 6622fa76b0e7790115ca858d5282eca7.png

Worked example 19: Sine function

  1. Sketch the following functions on the same set of axes for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\).
    1. \(y_1 = \sin \theta\)
    2. \(y_2 = \sin (\theta - \text{30}\text{°})\)

  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 \(y = \sin (\theta + p)\)

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

Complete a table of values

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

Sketch the sine graphs

b6140efff45cebda73c2a619540a6289.png

Complete the table

\(y_1 = \sin \theta\) \(y_2 = \sin (\theta - \text{30}\text{°})\)
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 \((-\text{270}\text{°};1)\) and \((\text{90}\text{°};1)\) \((-\text{240}\text{°};1)\) and \((\text{120}\text{°};1)\)
minimum turning points \((-\text{90}\text{°};-1)\) and \((\text{270}\text{°};-1)\) \((-\text{60}\text{°};-1)\) and \((\text{300}\text{°};-1)\)
\(y\)-intercept(s) \((\text{0}\text{°};0)\) \((\text{0}\text{°};-\frac{1}{2})\)
\(x\)-intercept(s) \((-\text{360}\text{°};0)\), \((-\text{180}\text{°};0)\), \((\text{0}\text{°};0)\), \((\text{180}\text{°};0)\) and \((\text{360}\text{°};0)\) \((-\text{330}\text{°};0)\), \((-\text{150}\text{°};0)\), \((\text{30}\text{°};0)\) and \((\text{210}\text{°};0)\)

Discovering the characteristics

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

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 \(f(\theta)\).

temp text

Sine functions of the form \(y = \sin (\theta + p)\)

Textbook Exercise 5.22

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

\(f(\theta) =\sin (\theta + \text{30}\text{°})\)

f4a9946872f7e284148a603e562771a4.png

For \(f(\theta) =\sin (\theta + \text{30}\text{°})\):

\begin{align*} \text{Period: } & \text{360}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{360}\text{°};\text{360}\text{°}] \\ \text{Range: } & [-1;1] \\ x\text{-intercepts: } & (-\text{210}\text{°};0); (-\text{30}\text{°};0); (\text{150}\text{°};0); (\text{330}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};\frac{1}{2}) \\ \text{Max. turning point: } & (-\text{300}\text{°};1); (\text{60}\text{°};1) \\ \text{Min. turning point: } & (-\text{120}\text{°};-1); (\text{240}\text{°};-1) \end{align*}

\(g(\theta) =\sin (\theta - \text{45}\text{°})\)

f4a9946872f7e284148a603e562771a4.png

For \(g(\theta) =\sin (\theta - \text{45}\text{°})\):

\begin{align*} \text{Period: } & \text{360}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{360}\text{°};\text{360}\text{°}] \\ \text{Range: } & [-1;1] \\ x\text{-intercepts: } & (-\text{315}\text{°};0); (-\text{135}\text{°};0); (\text{45}\text{°};0); (\text{225}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};\frac{1}{\sqrt{2}}) \\ \text{Max. turning point: } & (-\text{300}\text{°};1); (-\text{225}\text{°};1) ; (\text{135}\text{°};1) \\ \text{Min. turning point: } & (-\text{45}\text{°};-1); (\text{315}\text{°};-1) \end{align*}

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

54a7cd4ed1fbac0a6e33f20a1e87a38a.png

For \(h(\theta) =\sin (\theta + \text{60}\text{°})\):

\begin{align*} \text{Period: } & \text{360}\text{°} \\ \text{Amplitude: } & 1 \\ \text{Domain: } & [-\text{360}\text{°};\text{360}\text{°}] \\ \text{Range: } & [-1;1] \\ x\text{-intercepts: } & (-\text{240}\text{°};0); (-\text{60}\text{°};0); (\text{120}\text{°};0); (\text{300}\text{°};0) \\ y\text{-intercepts: } & (\text{0}\text{°};\frac{\sqrt{3}}{2}) \\ \text{Max. turning point: } & (-\text{300}\text{°};1); (-\text{330}\text{°};1) ; (\text{30}\text{°};1) \\ \text{Min. turning point: } & (-\text{150}\text{°};-1); (\text{210}\text{°};-1) \end{align*}

Sketching sine graphs (EMBH2)

Worked example 20: Sketching a sine graph

Sketch the graph of \(f(\theta) = \sin (\text{45}\text{°} - \theta)\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).

Examine the form of the equation

Write the equation in the form \(y = \sin (\theta + p)\).

\begin{align*} f(\theta) &= \sin (\text{45}\text{°} - \theta)\\ &= \sin (-\theta + \text{45}\text{°}) \\ &= \sin \left( -(\theta - \text{45}\text{°}) \right) \\ &= -\sin (\theta - \text{45}\text{°}) \end{align*}

To draw a graph of the above function, we know that the standard sine graph, \(y = \sin\theta\), must:

  • be reflected about the \(x\)-axis
  • be shifted to the right by \(\text{45}\text{°}\)

Complete a table of values

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

Plot the points and join with a smooth curve

c48bdaef4f3955e156bcc39c2c65dd9e.png

Period: \(\text{360}\text{°}\)

Amplitude: \(\text{1}\)

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

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

Maximum turning point: \((\text{315}\text{°};1)\)

Minimum turning point: \((\text{135}\text{°};-1)\)

\(y\)-intercepts: \((\text{0}\text{°};\text{0,71})\)

\(x\)-intercept: \((\text{45}\text{°};0) \text{ and } (\text{225}\text{°};0)\)

Worked example 21: Sketching a sine graph

Sketch the graph of \(f(\theta) = \sin (3\theta + \text{60}\text{°})\) for \(\text{0}\text{°} \leq \theta \leq \text{180}\text{°}\).

Examine the form of the equation

Write the equation in the form \(y = \sin k(\theta + p)\).

\begin{align*} f(\theta) &= \sin (3\theta + \text{60}\text{°})\\ &= \sin 3(\theta + \text{20}\text{°}) \end{align*}

To draw a graph of the above equation, the standard sine graph, \(y = \sin\theta\), must be changed in the following ways:

  • decrease the period by a factor of \(\text{3}\);
  • shift to the left by \(\text{20}\text{°}\).

Complete a table of values

θ \(\text{0}\)\(\text{°}\) \(\text{30}\)\(\text{°}\) \(\text{60}\)\(\text{°}\) \(\text{90}\)\(\text{°}\) \(\text{120}\)\(\text{°}\) \(\text{150}\)\(\text{°}\) \(\text{180}\)\(\text{°}\)
\(f(\theta)\) \(\text{0,87}\) \(\text{0,5}\) \(-\text{0,87}\) \(-\text{0,5}\) \(\text{0,87}\) \(\text{0,5}\) \(-\text{0,87}\)

Plot the points and join with a smooth curve

df7384a16d5d9bb9e2dcbb588d9ab470.png

Period: \(\text{120}\text{°}\)

Amplitude: \(\text{1}\)

Domain: \([\text{0}\text{°}; \text{180}\text{°}]\)

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

Maximum turning point: \((\text{10}\text{°}; 1) \text{ and } (\text{130}\text{°}; 1)\)

Minimum turning point: \((\text{70}\text{°}; -1)\)

\(y\)-intercept: \((\text{0}\text{°}; \text{0,87})\)

\(x\)-intercepts: \((\text{40}\text{°}; 0)\), \((\text{100}\text{°}; 0)\) and \((\text{160}\text{°}; 0)\)

The sine function

Textbook Exercise 5.23

Sketch the following graphs on separate axes:

\(y = 2 \sin \frac{\theta}{2}\) for \(-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}\)

55d67e42135e2672bbdd1392c518c70c.png

\(f(\theta) = \frac{1}{2} \sin (\theta - \text{45}\text{°})\) for \(-\text{90}\text{°} \leq \theta \leq \text{90}\text{°}\)

5b9dfaf0689281db46e3bf5013120ba0.png

\(y = \sin (\theta + \text{90}\text{°}) + 1\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)

099f1df4452603ac03b7efc2e0896d85.png

\(y = \sin (-\frac{3\theta}{2})\) for \(-\text{180}\text{°} \leq \theta \leq \text{180}\text{°}\)

072a75cd31101f2205ead2a0fc44d19d.png

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

3d799548e7b6929ae9e535841e4c071a.png

Given the graph of the function \(y = a \sin (\theta + p)\), determine the values of \(a\) and \(p\).

30c687451c5c4cebef7b24086f93a9e2.png

Can you describe this graph in terms of \(\cos \theta\)?

\(a = 2\); \(p = \text{90}\text{°} \therefore y = 2 \sin ( \theta + \text{90}\text{°})\) and \(y = 2 \cos \theta\)
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5.6 The cosine function