Q. 21 - Precalculus Enhanced with Graphing Utilities

Question

find the value of the remaining five trigonometric functions of

$$\begin{gathered} \tan \vartheta =\frac{-12}{5} \\ \pi /2<\vartheta <\pi \end{gathered}$$

Step-by-Step Solution

Answer

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

$$\begin{gathered} \cos \vartheta = -\frac{5}{13} \\ \sin \vartheta =\frac{12}{13} \\ \cos ec \vartheta =\frac{13}{12} \\ sec\vartheta =-\frac{13}{5} \\ cot \vartheta =-\frac{5}{12} \end{gathered}$$

Step 1. Given information

$$\begin{gathered} \tan \vartheta =\frac{-12}{5} \\ \pi /2<\vartheta <\pi \end{gathered}$$

Step 2. Solving for s e c &#977; &#160; &#38; &#160; cos &#977;

We know that ;

${\tan }^2\vartheta +1 =se{c}^2\vartheta$

Using this equation:

$$\begin{gathered} sec\vartheta =\sqrt{1+{\tan }^2\vartheta } \\ sec\vartheta =\sqrt{1+{(-\frac{12}{5})}^2} \\ sec\vartheta =\sqrt{1+\frac{144}{25}} \\ sec\vartheta =\sqrt{\frac{169}{25}} \\ sec\vartheta =\frac{13}{5} \end{gathered}$$

Since theta is in second quadrant so;

$sec\vartheta =-\frac{13}{5}$

Step 3. Solving for cos &#977; &#160; , &#160; sin &#977; &#160; , &#160; c o t &#977; &#160; , cos e c &#977;

We know that :

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

Using this :

$$\begin{gathered} \cos \vartheta \frac{-5}{13} \\ \sin \vartheta = \cos \vartheta \times \tan \vartheta \\ \sin \vartheta =\frac{12}{13} \\ cot\vartheta = \frac{-5}{12} \\ \cos ec\vartheta = \frac{13}{12} \end{gathered}$$