To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives

\[ \left(\dfrac{3\pi}{5}\right) \cdot \left(\dfrac {180^\circ} {\pi}\right) = 108^\circ. \]

Therefore $\dfrac{3\pi}{5}$ is equivalent to $108^\circ,$ which lies in the $2$nd quadrant.

To express $\sin\left(\dfrac{3\pi}{5}\right)$ in terms of $\sin\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{3\pi}{5}$ is in the $2$nd quadrant, the reference angle $\theta_R$ is

\begin{align*} \theta_R &= \pi - \theta\\[5pt] &= \pi - \dfrac{3\pi}{5} \\[5pt] &= \dfrac{2\pi}{5}. \end{align*}

Step 2: The given ratio is $\sin\left(\dfrac{3\pi}{5}\right)$, and therefore we're interested in $\sin\theta_R = \sin\left(\dfrac{2\pi}{5}\right).$

Step 3: The ratio $\sin\left(\dfrac{3\pi}{5}\right)$ must be positive because the sine ratio is always positive in the $2$nd quadrant. Therefore,

\[ \sin\left(\dfrac{3\pi}{5}\right) = \sin\left(\dfrac{2\pi}{5}\right). \]

To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives

\[ \left(\dfrac{4\pi} {3}\right) \cdot \left(\dfrac {180^\circ} {\pi}\right) = 240^\circ. \]

Therefore $\dfrac{4\pi} {3}$ is equivalent to $240^\circ,$ which lies in the $3$rd quadrant.

To express $\sin\left(\dfrac{4\pi}{3}\right)$ in terms of $\sin\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{4\pi}{3}$ is in the $3$rd quadrant, the reference angle $\theta_R$ is

\begin{align*} \theta_R &= \theta- \pi\\[5pt] &= \dfrac{4\pi}{3}-\pi \\[5pt] &= \dfrac{\pi}{3}. \end{align*}

Step 2: The given ratio is $\sin\left(\dfrac{4\pi}{3}\right)$, and therefore we're interested in $\sin\theta_R = \sin\left(\dfrac{\pi}{3}\right).$

Step 3: The ratio $\sin\left(\dfrac{4\pi}{3}\right)$ must be negative because the sine ratio is always negative in the $3$rd quadrant. Therefore,

\[ \sin\left(\dfrac{4\pi}{3}\right) = -\sin\left(\dfrac{\pi}{3}\right). \]

To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives

\[ \left(\dfrac{7\pi}{4}\right) \cdot \left(\dfrac {180^\circ} {\pi}\right) = 315^\circ. \]

Therefore $\dfrac{7\pi}{4}$ is equivalent to $315^\circ,$ which lies in the $4$th quadrant.

To express $\cos\left(\dfrac{7\pi}{4}\right)$ in terms of $\cos\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{7\pi}{4}$ is in the $4$th quadrant, the reference angle $\theta_R$ is

\begin{align*} \theta_R &= 2\pi - \theta\\[5pt] &= 2\pi - \dfrac{7\pi}{4}\\[5pt] &= \dfrac{\pi}{4}. \end{align*}

Step 2: The given ratio is $\cos\left(\dfrac{7\pi}{4}\right)$, and therefore we're interested in $\cos\theta_R = \cos\left(\dfrac{\pi}{4}\right).$

Step 3: The ratio $\cos\left(\dfrac{7\pi}{4}\right)$ must be positive because the cosine ratio is always positive in the $4$th quadrant. Therefore,

\[ \cos\left(\dfrac{7\pi}{4}\right) = \cos\left(\dfrac{\pi}{4}\right). \]

To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives

\[ \left(\dfrac{4\pi} {3}\right) \cdot \left(\dfrac {180^\circ} {\pi}\right) = 240^\circ. \]

Therefore $\dfrac{4\pi} {3}$ is equivalent to $240^\circ,$ which lies in the $3$rd quadrant.

To express $\cos\left(\dfrac{4\pi}{3}\right)$ in terms of $\cos\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{4\pi}{3}$ is in the $3$rd quadrant, the reference angle $\theta_R$ is

\begin{align*} \theta_R &= \theta-\pi\\[5pt] &= \dfrac{4\pi}{3}-\pi\\[5pt] &= \dfrac{\pi}{3}. \end{align*}

Step 2: The given ratio is $\cos\left(\dfrac{4\pi}{3}\right)$, and therefore we're interested in $\cos\theta_R = \cos\left(\dfrac{\pi}{3}\right).$

Step 3: The ratio $\cos\left(\dfrac{4\pi}{3}\right)$ must be negative because the cosine ratio is always negative in the $3$rd quadrant. Therefore,

\[ \cos\left(\dfrac{4\pi}{3}\right) = -\cos\left(\dfrac{\pi}{3}\right). \]

To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives:

\[ \left(\dfrac{7\pi} {6}\right) \cdot \left(\dfrac {180^\circ} {\pi}\right) = 210^\circ \]

Therefore $\dfrac{7\pi} {6}$ is equivalent to $210^\circ,$ which lies in the $3\textrm{rd}$ quadrant.

To express $\tan\left(\dfrac{7\pi}{6}\right)$ in terms of $\tan\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{7\pi}{6}$ is in the $3$rd quadrant, the reference angle $\theta_R$ is

\begin{align*} \theta_R &= \theta-\pi\\[5pt] &= \dfrac{7\pi}{6}-\pi\\[5pt] &= \dfrac{\pi}{6}. \end{align*}

Step 2: The given ratio is $\tan\left(\dfrac{7\pi}{6}\right)$, and therefore we're interested in $\tan\theta_R = \tan\left(\dfrac{\pi}{6}\right).$

Step 3: The ratio $\tan\left(\dfrac{7\pi}{6}\right)$ must be positive because the tangent ratio is always positive in the $3$rd quadrant. Therefore,

\[ \tan\left(\dfrac{7\pi}{6}\right) = \tan\left(\dfrac{\pi}{6}\right). \]

To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives:

\[ \left(\dfrac{5\pi} {3}\right) \cdot \left(\dfrac {180^\circ} {\pi}\right) = 300^\circ \]

Therefore $\dfrac{5\pi} {3}$ is equivalent to $300^\circ,$ which lies in the $4\textrm{th}$ quadrant.

To express $\tan\left(\dfrac{5\pi}{3}\right)$ in terms of $\tan\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{5\pi}{3}$ is in the $4$th quadrant, the reference angle $\theta_R$ is

\begin{align*} \theta_R &= 2\pi - \theta\\[5pt] &= 2\pi - \dfrac{5\pi}{3}\\[5pt] &= \dfrac{\pi}{3}. \end{align*}

Step 2: The given ratio is $\tan\left(\dfrac{5\pi}{3}\right)$, and therefore we're interested in $\tan\theta_R = \tan\left(\dfrac{\pi}{3}\right).$

Step 3: The ratio $\tan\left(\dfrac{5\pi}{3}\right)$ must be negative because the tangent ratio is always negative in the $4$th quadrant. Therefore,

\[ \tan\left(\dfrac{5\pi}{3}\right) = -\tan\left(\dfrac{\pi}{3}\right). \]

To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives \[ \left(\dfrac{9\pi}{5}\right) \cdot \left(\dfrac{180^\circ}{\pi}\right) = 324^\circ. \]

Therefore $\dfrac{9\pi}{5}$ is equivalent to $324^\circ,$ which lies in the $4\textrm{th}$ quadrant.

To express $\sec\left(\dfrac{9\pi}{5}\right)$ in terms of $\sec\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{9\pi}{5}$ is in the $4$th quadrant, the reference angle $\theta_R$ is \begin{align*} \theta_R & = 2\pi - \theta\\[5pt] &= 2\pi - \dfrac{9\pi}{5}\\[5pt] & = \dfrac{\pi}{5}. \end{align*}

Step 2: The given ratio is $\sec\left(\dfrac{9\pi}{5}\right)$, and therefore we're interested in $\sec\theta_R = \sec\left(\dfrac{\pi}{5}\right).$

Step 3: The ratio $\sec\left( \dfrac{9\pi}{5}\right)$ must be positive because cosine (and therefore secant) is always positive in the $4$th quadrant. Therefore, \[ \sec\left( \dfrac{9\pi}{5}\right) = \sec\left(\dfrac{\pi}{5}\right). \]

To help us to visualize the angle, we first convert the measure of the angle into degrees.

To convert the measure of an angle in radians to an equivalent measure in degrees, we multiply the measure in radians by $\dfrac{180^\circ}{\pi}.$ This gives

\[ \left(\dfrac{13\pi} {8}\right) \cdot \left(\dfrac {180^\circ} {\pi}\right) = 292.5^\circ. \]

Therefore $\dfrac{13\pi} {8}$ is equivalent to $292.5^\circ,$ which lies in the $4\textrm{th}$ quadrant.

To express $\csc\left(\dfrac{13\pi}{8}\right)$ in terms of $\csc\theta_R,$ we follow three steps:

1. Find its reference angle $\theta_R.$

2. Calculate the value of the function for $\theta_R.$

3. Determine whether the resulting value is positive or negative.

Step 1: Since $\theta = \dfrac{13\pi}{8}$ is in the $4$th quadrant, the reference angle $\theta_R$ is

\begin{align*} \theta_R &= 2\pi-\theta\\[5pt] &= 2\pi-\dfrac{13\pi}{8}\\[5pt] &= \dfrac{3\pi}{8}. \end{align*}

Step 2: The given ratio is $\csc\left(\dfrac{13\pi}{8}\right)$, and therefore we're interested in $\csc\theta_R = \csc\left(\dfrac{3\pi}{8}\right).$

Step 3: The ratio $\csc\left( \dfrac{13\pi}{8}\right)$ must be negative because sine (and therefore cosecant) is always negative in the $4$th quadrant. Therefore,

\[ \csc\left( \dfrac{13\pi}{8}\right) = -\csc\left(\dfrac{3\pi}{8}\right). \]