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?

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