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

The Sine 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

Do you need more Practice?

Siyavula Practice gives you access to unlimited questions with answers that help you learn. Practise anywhere, anytime, and on any device!

Sign up to practise now

Revision

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.png9489eeaa93dfc8a3ea08de3bbb12e149.png

\(k > 1\)

\(k < -1\)

aed1a16b6d01b9250823e94de6445009.png4c3ec29fdca6d6878e134da51a3c64d7.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 \((\text{0}\text{°};0)\).

Do you need more Practice?

Siyavula Practice gives you access to unlimited questions with answers that help you learn. Practise anywhere, anytime, and on any device!

Sign up to practise now

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

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.png6622fa76b0e7790115ca858d5282eca7.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 \(f(\theta)\).

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

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

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 \(y = 2 \cos \theta\)