Answer
The values of the remaining trigonometric function are,
$\cos t=\frac{\sqrt{21}}{7},\text{ }\tan t=\frac{2\sqrt{3}}{3},\text{ }\csc t=\frac{\sqrt{7}}{2}\text{, }\sec t=\frac{\sqrt{21}}{3},\text{ }\cot t=\frac{\sqrt{3}}{2}$
Work Step by Step
Consider the provided trigonometric functions:
$\sin t=\frac{2}{\sqrt{7}}$
According to Pythagorean Identities:
${{\sin }^{2}}t+{{\cos }^{2}}t=1$
Now, substitute the value of $\sin t $ in the equation ${{\sin }^{2}}t+{{\cos }^{2}}t=1$.
Therefore,
${{\left( \frac{2}{\sqrt{7}} \right)}^{2}}+{{\cos }^{2}}t=1$
The square of, ${{\left( \frac{2}{\sqrt{7}} \right)}^{2}}$ is $\left( \frac{4}{7} \right)$,
Therefore,
$\frac{4}{7}+{{\cos }^{2}}t=1$
Subtract, $\frac{4}{7}$ from the sides,
$\begin{align}
& -\frac{4}{7}+\frac{4}{7}+{{\cos }^{2}}t=1-\frac{4}{7} \\
& {{\cos }^{2}}t=1-\frac{4}{7}
\end{align}$
On further simplification,
$\begin{align}
& {{\cos }^{2}}t=1-\frac{4}{7} \\
& {{\cos }^{2}}t=\frac{3}{7} \\
\end{align}$
Take the square root on both sides,
$\cos t=\sqrt{\frac{3}{7}}$
This implies that
$\cos t=\sqrt{\frac{3}{7}}=\frac{\sqrt{21}}{7}$
The value of $\cos t $ is positive in the given interval $0\le t\le \frac{\pi }{4}$.
Hence, the value of $\cos t $ is $\frac{\sqrt{21}}{7}$.
Now, the value of $\tan t $ can be obtained by using the values of $\sin t $ and $\cos t $.
Therefore,
$\tan t=\frac{y}{x}$
Substitute the value of $\sin t $ and $\cos t $ in the function $\tan t=\frac{y}{x}$.
$\tan t=\frac{\frac{2}{\sqrt{7}}}{\sqrt{\frac{3}{7}}}$
This implies that
$\tan t=\frac{2\sqrt{3}}{3}$
Hence, the value of $\tan t $ is $\frac{2\sqrt{3}}{\sqrt{3}}$
Now, for $\csc t $ as:
$\csc t=\frac{1}{\sin t}$
Substitute, the values of $\sin t $ in $\csc t=\frac{1}{\sin t}$.
$\csc t=\frac{1}{\frac{2}{\sqrt{7}}}$
This implies that
$\csc t=\frac{\sqrt{7}}{2}$
Hence, the value of $\csc t $ is $\frac{\sqrt{7}}{2}$
Now, for $\sec t $ as:
$\sec t=\frac{1}{\cos t}$
Substitute, the value of $\cos t $ in $\sec t=\frac{1}{\cos t}$,
$\begin{align}
& \sec t=\frac{1}{\frac{\sqrt{21}}{7}} \\
& =\frac{7}{\sqrt{21}}
\end{align}$
This implies that
$\sec t=\frac{\sqrt{21}}{3}$
Hence, the value of $\sec t $ is $\frac{\sqrt{21}}{3}$ .
Now, for $\cot t $ as:
$\cot t=\frac{x}{y}$
Substitute the value of $\sin t $ and $\cos t $ in the function $\cot t=\frac{x}{y}$.
$\cot t=\frac{\frac{2}{\sqrt{7}}}{\frac{\sqrt{21}}{7}}$
This implies that
$\cot t=\frac{\sqrt{3}}{2}$
Hence, the value of $\cot t $ is $\frac{\sqrt{3}}{2}$ .
