Question

How do you find the values of the six trigonometric functions given $\tan \theta$ is undefined and $\pi \leqslant \theta \leqslant 2\pi$?

Answer 1

We have to find the values of the six trigonometric functions given $\tan \theta$ is undefined and $\pi \leqslant \theta \leqslant 2\pi$. For this, first find the angle $\theta$ for which $\tan \theta$ is undefined and $\pi \leqslant \theta \leqslant 2\pi$. Then, find other five trigonometric functions at this angle $\theta$ using trigonometric values and identities.

Formula used: $\sin \left( {\dfrac{{3\pi }}{2}} \right) = - 1$
$\cos \left( {\dfrac{{3\pi }}{2}} \right) = 0$
$\cos \left( \theta \right) \times \sec \left( \theta \right) = 1$
$\sin \left( \theta \right) \times \cos ec\left( \theta \right) = 1$
$\tan \left( \theta \right) \times \cot \left( \theta \right) = 1$

Complete step-by-step solution:
We have to find the values of the six trigonometric functions given $\tan \theta$ is undefined and $\pi \leqslant \theta \leqslant 2\pi$.
So, first we have to find the angle $\theta$ for which $\tan \theta$ is undefined and $\pi \leqslant \theta \leqslant 2\pi$.
We know that $\tan \theta$ is undefined for $\theta = \dfrac{\pi }{2}$.
But $\theta \in \left[ {\pi ,2\pi } \right]$.
So, we can find the angle $\theta$ by adding $\dfrac{\pi }{2}$ to $\pi$ or subtracting $\dfrac{\pi }{2}$ from $2\pi$.
$\theta = \pi + \dfrac{\pi }{2}$
$\Rightarrow \theta = \dfrac{{3\pi }}{2}$
Now, we will find the other five trigonometric functions on $\theta = \dfrac{{3\pi }}{2}$.
Since, $\sin \left( {\dfrac{{3\pi }}{2}} \right) = - 1$.
$\Rightarrow \sin \left( \theta \right) = \sin \left( {\dfrac{{3\pi }}{2}} \right) = - 1$
Since, $\cos \left( {\dfrac{{3\pi }}{2}} \right) = 0$.
$\Rightarrow \cos \left( \theta \right) = \cos \left( {\dfrac{{3\pi }}{2}} \right) = 0$
Now, using trigonometry identity $\cos \left( \theta \right) \times \sec \left( \theta \right) = 1$, we get
$\Rightarrow \sec \left( \theta \right) = \dfrac{1}{{\cos \left( \theta \right)}} = \dfrac{1}{0}$ = undefined
Now, using trigonometry identity $\sin \left( \theta \right) \times \cos ec\left( \theta \right) = 1$, we get
$\Rightarrow \cos ec\left( \theta \right) = \dfrac{1}{{\sin \left( \theta \right)}} = \dfrac{1}{{ - 1}} = - 1$
Now, using trigonometry identity $\tan \left( \theta \right) \times \cot \left( \theta \right) = 1$, we get
$\cot \left( \theta \right) = \dfrac{1}{{\tan \left( \theta \right)}} = \dfrac{0}{1} = 0$
Final solution: Therefore, $\sin \left( \theta \right) = - 1$, $\cos \left( \theta \right) = 0$, $\sec \left( \theta \right) =$ undefined, $\cos ec\left( \theta \right) = - 1$ and $\cot \left( \theta \right) = 0$.
Additional information: Trigonometric identity: An equation involving trigonometric ratios of an angle $\theta$ (say) is said to be a trigonometric identity if it is satisfied for all values of $\theta$ for which the given trigonometric ratios are defined.
For example, ${\cos ^2}\theta - \dfrac{1}{2}\cos \theta = \cos \theta \left( {\cos \theta - \dfrac{1}{2}} \right)$ is a trigonometric identity, whereas $\cos \theta \left( {\cos \theta - \dfrac{1}{2}} \right) = 0$ is an equation.
Also, $\sec \theta = \dfrac{1}{{\cos \theta }}$ is a trigonometric identity, because it holds for all values of $\theta$ except for which $\cos \theta = 0$. For $\cos \theta = 0$, $\sec \theta$ is not defined.

Note: We can directly find the trigonometric functions using trigonometric identities:
${\sin ^2}\theta + {\cos ^2}\theta = 1$.........…(1)
${\sec ^2}\theta - {\tan ^2}\theta = 1$.........…(2)
$\cos e{c^2}\theta - {\cot ^2}\theta = 1$………...(3)
So, first we can determine $\sec \theta$ using trigonometry identity (2).
${\sec ^2}\theta = 1 + {\tan ^2}\theta$
$\Rightarrow \sec \theta = \pm \sqrt {1 + {{\tan }^2}\theta }$
Since, $\tan \theta$ is undefined. So, $\tan \theta = \dfrac{1}{0}$.
$\Rightarrow \sec \left( \theta \right) =$ undefined
Now, using trigonometry identity $\cos \left( \theta \right) \times \sec \left( \theta \right) = 1$, we get
$\cos \left( \theta \right) = \dfrac{1}{{\sec \left( \theta \right)}} = \dfrac{0}{1} = 0$
Now, we can determine $\sin \theta$ using trigonometry identity (1).
${\sin ^2}\theta + {\cos ^2}\theta = 1$
$\Rightarrow \sin \theta = \pm \sqrt {1 - {{\cos }^2}\theta }$
$\Rightarrow \sin \theta = \pm \sqrt {1 - 0}$
$\Rightarrow \sin \theta = \pm 1$
Since, $\pi \leqslant \theta \leqslant 2\pi$.
$\Rightarrow \sin \theta = - 1$
Now, using trigonometry identity $\sin \left( \theta \right) \times \cos ec\left( \theta \right) = 1$, we get
$\Rightarrow \cos ec\left( \theta \right) = \dfrac{1}{{\sin \left( \theta \right)}} = \dfrac{1}{{ - 1}} = - 1$
Now, using trigonometry identity $\tan \left( \theta \right) \times \cot \left( \theta \right) = 1$, we get
$\cot \left( \theta \right) = \dfrac{1}{{\tan \left( \theta \right)}} = \dfrac{0}{1} = 0$
Therefore, $\sin \left( \theta \right) = - 1$, $\cos \left( \theta \right) = 0$, $\sec \left( \theta \right) =$ undefined, $\cos ec\left( \theta \right) = - 1$ and $\cot \left( \theta \right) = 0$.