Question

find the value of the remaining five trigonometric functions of $\vartheta$

$\sin \vartheta =\frac{5}{7}$, $\vartheta$ in Second Quadrant.

Short answer

the value of the remaining five trigonometric functions of $\vartheta$are:

$$\begin{gathered} \cos \vartheta =-\frac{2\sqrt{6}}{7} \\ \tan \vartheta =-\frac{5}{2\sqrt{6}} \\ \cos ec\vartheta =\frac{7}{5} \\ sec\vartheta =-\frac{7}{2\sqrt{6}} \\ cot\vartheta =-\frac{2\sqrt{6}}{5} \end{gathered}$$

Step-by-step solution

Step 1. Given information

$\sin \vartheta =\frac{5}{7}$,$\vartheta$ in Second Quadrant.

Step 2. Solving for  cosϑ

We know ;

${\sin }^2\vartheta +{\cos }^2\vartheta =1$

Using this equation :

$$\begin{gathered} \cos \vartheta =\sqrt{1-{\sin }^2\vartheta } \\ \cos \vartheta =\sqrt{1-{\left(\frac{5}{7}\right)}^2} \\ \cos \vartheta =\sqrt{1-\frac{25}{49}} \\ \cos \vartheta =\sqrt{\frac{24}{49}} \\ \cos \vartheta =\frac{2\sqrt{6}}{7} \end{gathered}$$

Since theta is in second quadrant so,

$\cos \vartheta =-\frac{2\sqrt{6}}{7}$

Step 3. Solving forcotϑ , secϑ , cosecϑ

We know ;

$$\begin{gathered} \cos ec\vartheta =\frac{1}{\sin \vartheta } \\ sec\vartheta =\frac{1}{\cos \vartheta } \\ cot\vartheta =\frac{1}{\tan \vartheta } \\ \tan \vartheta =\frac{\sin \vartheta }{\cos \vartheta } \end{gathered}$$

Using the above identities :

$$\begin{gathered} \tan \vartheta =-\frac{5}{2\sqrt{6}} \\ \cos ec\vartheta =\frac{7}{5} \\ sec\vartheta =-\frac{7}{2\sqrt{6}} \\ cot\vartheta =-\frac{2\sqrt{6}}{5} \end{gathered}$$