Question

Find the trigonometric ratios equivalent to $\csc{{45}^{\circ }}$ .
(This question has multiple correct options)
A. $\sqrt{2}$
B. $\sec \,{{45}^{\circ }}$
C. $\dfrac{1}{\sin {{45}^{\circ }}}$
D. $1+{{\cot }^{2}}{{45}^{\circ }}$

Answer 1

Hint: Here we have given a trigonometric function with an angle and we have to find what all values from the option given are equal to. Firstly we will simplify the trigonometric value given by the help of sine function value and check which of the options is having that value. Then we will check what other values give the same answer and get our desired answer.

Complete step-by-step solution:
The trigonometric function is given as follows:
$\csc {{45}^{\circ }}$
Now as we know the cosecant function is inverse of sine function so we can write,
$\sin {{45}^{\circ }}=\dfrac{1}{\csc {{45}^{\circ }}}$
\[\Rightarrow \csc {{45}^{\circ }}=\dfrac{1}{\sin {{45}^{\circ }}}\]…..$\left( 1 \right)$
Now as we know $\sin {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}$ so substitute it above and solve as follows:
$\csc {{45}^{\circ }}=\dfrac{1}{\dfrac{1}{\sqrt{2}}}$
$\Rightarrow \csc {{45}^{\circ }}=\sqrt{2}$…..$\left( 2 \right)$
So from equation (1) and (2) $\csc {{45}^{\circ }}$ is equal to $\sqrt{2}$ and $\dfrac{1}{\sin {{45}^{\circ }}}$ .
Hence the correct option is (A) and (C).

Note: As the value of sine, cosine and tangent function are known which one should learn at the starting of the topic trigonometry so whenever we have to find the value of any trigonometry function at an angle we convert the value in sine, cosine or tangent function as the calculation becomes easy. In this question we used the inverse relation between sine and cosecant because it is the simplest identity to find the cosecant values. There are various trigonometric identities that are used to solve more complicated problems such as the Pythagorean identity, Sum and Difference identity, Sine law and Cosine law etc. Trigonometry is a very important and widely used branch of mathematics it deals with the relation between the sides and the angles of a right-angle triangle