Question

How do you find the 6 trigonometric functions for 0 degrees?

Answer 1

Hint:
In this question, we are given the value of degree i.e., 0 degrees and we are asked to find the all other trigonometric ratio. First we will know the basic trigonometric functions, i.e., sin, cos and Now we have the value of $\sin \theta $and $\cos \theta $, by these values $\tan \theta $ can be calculated, and other trigonometric ratios can be calculated by finding the reciprocal of these trigonometric ratios.

Complete step by step solution:
The angles which lie between ${0^{\circ}}$ and ${90^{\circ}}$ are said to lie in the first quadrant. The angles between ${90^{\circ}}$ and ${180^{\circ}}$ are in the second quadrant, angles between ${180^{\circ}}$ and ${270^{\circ}}$ are in the third quadrant and angles between ${270^{\circ}}$ and ${360^{\circ}}$ are in the fourth quadrant.
In the first quadrant, the values for sin, cos and tan are positive.
In the second quadrant, the values for sin are positive only.
In the third quadrant, the values for tan are positive only.
In the fourth quadrant, the values for cos are positive only.
Now given degree is 0 ,
So, we know that $\sin {0^{\circ}} = 0$,
And $\cos {0^{\circ}} = 1$,
Now using the trigonometric identities we get,
$ \Rightarrow \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$,
Now substituting the value of $\sin \theta $ and $\cos \theta $ we get,
Here $\theta = {0^{\circ}}$,
$ \Rightarrow \tan {0^{\circ}} = \dfrac{0}{1}$,
Now simplifying we get,
$ \Rightarrow \tan {0^{\circ}} = 0$,
Now as $\cot \theta $ is the inverse of $\tan \theta $, we get,
$ \Rightarrow \cot \theta = \dfrac{1}{{\tan \theta }}$,
Now substituting the value of$\tan {0^{\circ}} = 0$, we get,
$ \Rightarrow \cot {0^{\circ}} = \dfrac{1}{0}$,
Now simplifying we get,
$ \Rightarrow \cot {0^0} = \infty $,
Now we know that $\csc \theta $ is the inverse of $\sin \theta $, we get,
$ \Rightarrow \csc \theta = \dfrac{1}{{\sin \theta }}$,
Now substituting the value of $\sin {0^{\circ}} = 0$, we get,
$ \Rightarrow \csc {0^{\circ}} = \dfrac{1}{0}$,
Now simplifying we get,
$\csc {0^{\circ}} = \infty $$ \Rightarrow \csc {0^{\circ}} = \infty $,
And we know that $\sec \theta $ is the inverse of $\cos \theta $ we get,
$ \Rightarrow \sec \theta = \dfrac{1}{{\cos \theta }}$,
Now substituting the value of $\cos {0^{\circ}} = 1$, we get,
$ \Rightarrow \sec {0^{\circ}} = \dfrac{1}{1}$,
Now simplifying we get,
$ \Rightarrow \sec {0^{\circ}} = 1$.
The value of trigonometric ratios are,$\cos {0^{\circ}} = 1$,$\sin {0^{\circ}} = 0$,$\tan {0^{\circ}} = 0$,$\cot {0^{\circ}} = \infty $,$\sec {0^{\circ}} = 1$,$\csc {0^{\circ}} = \infty $.

$\therefore $ The value of trigonometric ratios are $\cos {0^{\circ}} = 1$, $\sin {0^{\circ}} = 0$, $\tan {0^{\circ}} = 0$,$\cot {0^{\circ}} = \infty $,$\sec {0^{\circ}} = 1$, $\csc {0^{\circ}} = \infty $.

Note:
Most of the trigonometry calculations are done by using trigonometric ratios. There are 6 trigonometric ratios present in trigonometry. Every other important trigonometry formula is derived with the help of these ratios.
The 6 important ratios named as sin, cos, tan, sec, cot, sec. Sin and cos are fundamental or basic ratios whereas Tan, sec, cot, and csc are derived functions.