Question

What trigonometric ratios of angles from 0° to 90° are equal to the value ‘0’?

Answer 1

Hint – In order to find which trigonometric function with angles from 0° to 90° are equal to value 0, we write the values of all the trigonometric functions from the angles 0° to 90° and pick the function at an angle where it is zero.

Complete step-by-step answer:
The trigonometric table of all function with values of angles between 0° to 90° are shown below:

$\begin{array}{*{20}{c}}
  {{\text{Angle}}}&{{\text{Sin}}}&{{\text{Cos}}}&{{\text{Tan}}}&{{\text{Cot}}}&{{\text{Sec}}}&{{\text{Cosec}}} \\
  {0^\circ }&0&1&0&{{\text{undefined}}}&1&{{\text{undefined}}} \\
  {30^\circ }&{\dfrac{1}{2}}&{\dfrac{{\sqrt 3 }}{2}}&{\dfrac{1}{{\sqrt 3 }}}&{\sqrt 3 }&{\dfrac{2}{{\sqrt 3 }}}&2 \\
  {45^\circ }&{\dfrac{1}{{\sqrt 2 }}}&{\dfrac{1}{{\sqrt 2 }}}&1&1&{\sqrt 2 }&{\sqrt 2 } \\
  {60^\circ }&{\dfrac{{\sqrt 3 }}{2}}&{\dfrac{1}{2}}&{\sqrt 3 }&{\dfrac{1}{{\sqrt 3 }}}&2&{\dfrac{2}{{\sqrt 3 }}} \\
  {90^\circ }&1&0&{{\text{undefined}}}&0&{{\text{undefined}}}&1
\end{array}$

Hence from the table of trigonometric functions in between the angles 0° to 90°, we observe the value 0 occurring at the following function values –
Sin 0° = 0
Cos 90° = 0
Tan 0° = 0
Cot 90° = 0

Note – In order to solve this type of question the key is to accurately remember the values of all trigonometric functions from the angles 0° to 90°.
Otherwise, there is an easy way to memorize the trigonometric table, which is by memorizing the values of Sin and Cos functions and deriving the values of other trigonometric functions by expressing them in terms of Sin and Cos functions, as follows
${\text{Tan }}\theta {\text{ = }}\dfrac{{{\text{Sin }}\theta }}{{{\text{Cos }}\theta }}$
$
  {\text{Cot }}\theta {\text{ = }}\dfrac{{{\text{Cos }}\theta }}{{{\text{Sin }}\theta }} \\
  {\text{Cosec }}\theta {\text{ = }}\dfrac{1}{{{\text{Sin }}\theta }} \\
  {\text{Sec }}\theta {\text{ = }}\dfrac{1}{{{\text{Cos }}\theta }} \\
$