Question

Find the values of the six trigonometric functions of the function value constraint $\sin \left(\theta \right)=0$, $\sec \left(\theta \right)=-1$.

Answer 1

Find the value of six trigonometric functions using the given functions:

The given functions are:

1. $\sin \left(\theta \right)=0$
2. $\sec \left(\theta \right)=-1$

Use the basic trigonometric identity.

$\sec \left(\theta \right)=-1$ [Given]

$\Rightarrow$ $\frac{1}{\cos \left(\theta \right)}=-1$ $\left[\because \sec \left(x\right)=\frac{1}{\cos \left(x\right)}\right]$

$\Rightarrow$ $\cos \left(\theta \right)=-1$

$\tan \left(\theta \right)=\frac{\sin \left(\theta \right)}{\cos \left(\theta \right)}$ $\left[\because \tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}\right]$

$=\frac{0}{-1}$ [Put $\sin \left(\theta \right)=0$ and $\cos \left(\theta \right)=-1$ ]

$=0$ [Use $\frac{0}{a}=0,a\ne 0$ ]

$\therefore$ $\tan \left(\theta \right)=0$

$\cot \left(\theta \right)=\frac{1}{\tan \left(\theta \right)}$ $\left[\because \cot \left(x\right)=\frac{1}{\tan \left(x\right)}\right]$

$=\frac{1}{0}$ [$\because$ $\frac{a}{0}$ is undefined]

$=$undefined

$\therefore$ $\cot \left(\theta \right)=$ undefined

$cosec\left(\theta \right)=\frac{1}{\sin \left(\theta \right)}$ $\left[\because \csc \left(x\right)=\frac{1}{\sin \left(x\right)}\right]$

$=\frac{1}{0}$ [Put $\sin \left(\theta \right)=0$]

$=$undefined [ $\because$ $\frac{a}{0}$ is undefined]

$\therefore$ $cosec\left(\theta \right)=$undefined

Hence, the values of the six trigonometric functions for $\theta$ are

$\sin \left(\theta \right)=0$

$\cos \left(\theta \right)=-1$

$\tan \left(\theta \right)=0$

$cosec\left(\theta \right)=$Undefined

$\sec \left(\theta \right)=-1$

$\cot \left(\theta \right)=$Undefined.