Shown the following graph of the following form: \(y = a \sin{\theta} + q\) where **Point
A** is at \((180^{\circ};\text{1,5})\), and **Point B** is at
\((90^{\circ};3)\), find the values of \(a\) and \(q\).

To find \(q\) we note that \(q\) shifts the graph up or down. To determine \(q\) we can look at any
point on the graph. For instance point \(A\) is at \((180^{\circ};\text{1,5})\). For an unshifted sine
graph point \(A\) would be at \((180^{\circ};0)\). For this graph we see that this point has been
shifted up by \(\text{1,5}\) or \(\frac{3}{2}\) spaces. Therefore \(q = \frac{3}{2}\).

To find \(a\) we note that the \(y\)-value at the middle (point \(A\)) is \(\text{1,5}\), while the
\(y\)-value at the top (point \(B\)) is 3. We can find the amplitude by working out the distance from
the top of the graph to the middle of the graph: \(3 - \text{1,5} = \text{1,5}\). Therefore \(a =
\frac{3}{2}\).

The complete equation for the graph shown in this question is \(y = \frac{3}{2}\sin \theta +
\frac{3}{2}\).

Therefore \(a = \frac{3}{2} \text{ and } q = \frac{3}{2}\)

Shown the following graph of the following form: \(y = a \sin{\theta} + q\) where **Point
A** is at \((270^{\circ};-\text{6})\), and **Point B** is at
\((90^{\circ};\text{2})\), determine the values of \(a\) and \(q\).

To find \(a\) we note that the \(y\)-value at the bottom (point \(A\)) is \(-\text{6}\), while the
\(y\)-value at the top (point \(B\)) is 2. We can find the amplitude by working out the distance from
the top of the graph to the bottom of the graph and then dividing this by 2 since this distance is
twice the amplitude: \(\frac{2 - (-6)}{2} = 4\). Therefore \(a = 4\).

To find \(q\) we note that \(q\) shifts the graph up or down. To determine \(q\) we can look at any
point on the graph. For instance point \(B\) is at \((90^{\circ};2)\). For an unshifted sine graph
with the same \(a\) value (i.e. \(4 \sin \theta\)) point \(B\) would be at \((90^{\circ};4)\). For
this graph we see that this point has been shifted down by \(\text{2}\) spaces. Therefore \(q = 2\).

The complete equation for the graph shown in this question is \(y = 4\sin \theta - 2\).

Therefore \(a = 4 \text{ and } q = -2\)

The graph below shows a trigonometric equation of the following form: \(y = a \cos{\theta} + q\). Two
points are shown on the graph: **Point A** at \((180^{\circ};-\text{1,5})\), and
**Point B**: \((0^{\circ};-\text{0,5})\). Calculate the values of \(a\) (the amplitude of
the graph) and \(q\) (the vertical shift of the graph).

To find \(a\) we note that the \(y\)-value at the bottom (point \(A\)) is \(-\text{1,5}\), while the
\(y\)-value at the top (point \(B\)) is \(-\text{0,5}\). We can find the amplitude by working out the
distance from the top of the graph to the bottom of the graph and then dividing this by 2 since this
distance is twice the amplitude: \(\frac{-\text{0,5} - (-\text{1,5})}{2} = \frac{1}{2}\). Therefore
\(a = \frac{1}{2}\).

To find \(q\) we note that \(q\) shifts the graph up or down. To determine \(q\) we can look at any
point on the graph. For instance point \(B\) is at \((0^{\circ};-\text{0,5})\). For an unshifted
cosine graph with the same \(a\) value (i.e. \(\frac{1}{2} \cos \theta\)) point \(B\) would be at
\((0^{\circ};\text{0,5})\). For this graph we see that this point has been shifted down by
\(\text{1}\) space. Therefore \(q = 1\).

The complete equation for the graph shown in this question is \(y = \frac{1}{2}\cos \theta - 1\).

Therefore \(a = \frac{1}{2}, \text{ and } q= -1\).

The graph below shows a trigonometric equation of the following form: \(y = a \cos{\theta} + q\). Two
points are shown on the graph: **Point A** at \((90^{\circ};\text{0,0})\), and
**Point B**: \((180^{\circ};-\text{0,5})\). Calculate the values of \(a\) (the amplitude
of the graph) and \(q\) (the vertical shift of the graph).

To find \(a\) we note that the \(y\)-value at the bottom (point \(B\)) is \(-\text{0,5}\), while the
\(y\)-value at the middle (point \(A\)) is \(\text{0}\). We can find the amplitude by working out the
distance from the top of the graph to the middle of the graph: \(\text{0} - (-\text{0,5}) =
\frac{1}{2}\). Therefore \(a = \frac{1}{2}\).

To find \(q\) we note that \(q\) shifts the graph up or down. To determine \(q\) we can look at any
point on the graph. For instance point \(A\) is at \((90^{\circ};\text{0})\). For an unshifted cosine
graph with the same \(a\) value (i.e. \(\frac{1}{2} \cos \theta\)) point \(B\) would be at
\((0^{\circ};\text{0})\). For this graph we see that this point has not been shifted. Therefore \(q =
0\).

The complete equation for the graph shown in this question is \(y = \frac{1}{2}\cos \theta\).

Therefore \(a = \frac{1}{2}, \text{ and } q= 0\).

On the graph below you see a tangent curve of the following form: \(y = a \tan{\theta} + q\). Two
points are labelled on the curve: **Point A** is at \(\left(0^{\circ}; \frac{1}{3}
\right)\), and **Point B** is at \(\left(45^{\circ}; \frac{10}{3} \right)\).

Calculate, or otherwise determine, the values of \(a\) and \(q\).

To find \(q\) we note that \(q\) shifts the graph up or down. To determine \(q\) we can look at any
point on the graph. For instance point \(A\) is at \(\left(0^{\circ};\frac{1}{3}\right)\). For an
unshifted tangent graph point \(A\) would be at \((0^{\circ};\text{0})\). For this graph we see that
this point been shifted upwards by a \(\frac{1}{3}\). Therefore \(q = \frac{1}{3}\).

To find \(a\) we can substitute point \(B\) into the equation for the tangent graph:

\begin{align*}
y & = a \tan{\theta} + \frac{1}{3} \\
\left( \frac{10}{3} \right) & = a \tan{45^{\circ}} + \frac{1}{3} \\
\frac{10}{3} & = a(1) + \frac{1}{3} \\
\frac{10}{3} - \frac{1}{3} & = a \\
3 & = a
\end{align*}

The complete equation is: \(y = 3 \tan{\theta} + \frac{1}{3}\).

Therefore \(a = 3 \text{ and } q = \frac{1}{3}\).

The graph below shows a tangent curve with an equation of the form \(y = a \tan{\theta} + q\). Two
points are labelled on the curve: **Point A** is at \((0^{\circ} ; 0)\), and
**Point B** is at \((45^{\circ} ; 1)\).

Find \(a\) and \(q\).

To find \(q\) we note that \(q\) shifts the graph up or down. To determine \(q\) we can look at any
point on the graph. For instance point \(A\) is at \((0^{\circ};0)\). For an unshifted tangent graph
point \(A\) would be at \((0^{\circ};\text{0})\). For this graph we see that the graph has not been
shifted. Therefore \(q = 0\).

To find \(a\) we can substitute point \(B\) into the equation for the tangent graph:

\begin{align*}
y & = a \tan{\theta} \\
1 & = a \tan{45^{\circ}} \\
1 & = a(1) \\
1 & = a
\end{align*}

The complete equation is: \(y = \tan{\theta}\).

Therefore \(a = 1 \text{ and } q = 0\).

The graph below shows functions \(f(x)\) and \(g(x)\)

What is the equation for \(g(x)\)?

\(g(x) = 4\sin{\theta}\)

With the assistance of the table below sketch the three functions on the same set of axes.

\[\begin{array}{| l | l | l | l | l | l | l | l | l | l | }
\hline
\theta & 0^{\circ} & 45^{\circ} & 90^{\circ} & 135^{\circ} & 180^{\circ} &
225^{\circ} & 270^{\circ} & 315^{\circ} & 360^{\circ} \\
\hline
\tan{\theta} & 0 & 1 & \text{undefined} & -1 & 0 & 1 & \text{undefined}
& -1 & 0 \\
\hline
3\tan{\theta} & 0 & 3 & \text{undefined} & -3 & 0 & 3 & \text{undefined}
& -3 & 0 \\
\hline
\frac{1}{2}\tan{\theta} & 0 & \frac{1}{2} & \text{undefined} & -\frac{1}{2} & 0
& \frac{1}{2} & \text{undefined} & -\frac{1}{2} & 0 \\
\hline
\end{array}\]

We are given a table with values and so we plot each of these points and join them with a smooth
curve.

With the assistance of the table below sketch the three functions on the same set of axes.

\[\begin{array}{| l | l | l | l | l | l | }
\hline
\theta & 0^{\circ} & 90^{\circ} & 180^{\circ} & 270^{\circ} & 360^{\circ} \\
\hline
\cos{\theta} - 2 & -1 & -2 & -3 &-2 & -1 \\
\hline
\cos{\theta} + 4 & 5 & 4 & 2 & 4 & 5 \\
\hline
\cos{\theta} + 2 & 3 & 2 & 1 & 2 & 3 \\
\hline
\end{array}\]

We are given a table with values and so we plot each of these points and join them with a smooth
curve.

State the coordinates at \(E\) and the domain and range of the function in the interval shown.

\(E(180^{\circ};1) \text{ , range } y \in \mathbb{R} \text{ and domain } 0 \le \theta \le 360, x \neq
90, x\neq 270\)

For which values of \(\theta\) is the function increasing, in the interval shown?

\(90^{\circ} < \theta < 270^{\circ}\)

For which values of \(\theta\) is the function negative, in the interval shown?

\(0^{\circ} < \theta < 210^{\circ} \text { and } 330^{\circ} < \theta < 360^{\circ}\)

For which values of \(\theta\) is the function positive, in the interval shown?

\(60^{\circ} < \theta < 300^{\circ}\)