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Question: How do you find the exact value of \[{\csc ^{ - 1}}\left( { - 1} \right)\]?...

How do you find the exact value of csc⁡−1(−1){\csc ^{ - 1}}\left( { - 1} \right)?

Explanation

Solution

Here, we will find the Trigonometric Value of the given Trigonometric Ratio. We will use the exponent rule and by using the trigonometric co-ratio where the trigonometric value is known and then we will find the value of the inverse trigonometric ratios. Thus, the Trigonometric Value of the given inverse Trigonometric Ratio is the required answer.

Formula Used:
We will use the following formula:
Exponent Rule:a−n=1an{a^{ - n}} = \dfrac{1}{{{a^n}}}
Trigonometric Co-ratio: csc⁡θ=1sin⁡θ\csc \theta = \dfrac{1}{{\sin \theta }}

Complete step by step solution:
We are given a Trigonometric ratio csc⁡−1(−1){\csc ^{ - 1}}\left( { - 1} \right).
Now, we will find the exact value of csc⁡−1(−1){\csc ^{ - 1}}\left( { - 1} \right).
Let θ\theta be the trigonometric angle.
⇒θ=csc⁡−1(−1)\Rightarrow \theta = {\csc ^{ - 1}}\left( { - 1} \right).
Exponent Rule:a−n=1an{a^{ - n}} = \dfrac{1}{{{a^n}}}
By using the Exponent Rule, we get
⇒θ=1csc⁡(−1)\Rightarrow \theta = \dfrac{1}{{\csc \left( { - 1} \right)}}.
We know that the trigonometric co-ratio of cosecant is sine i.e., csc⁡θ=1sin⁡θ\csc \theta = \dfrac{1}{{\sin \theta }}
By using the trigonometric co-ratio, we get
⇒θ=11sin⁡(−1)\Rightarrow \theta = \dfrac{1}{{\dfrac{1}{{\sin }}\left( { - 1} \right)}}.
By rewriting the equation, we get
⇒θ=sin⁡(−1)\Rightarrow \theta = \sin \left( { - 1} \right).
⇒θ=(−π2)\Rightarrow \theta = \left( { - \dfrac{\pi }{2}} \right).
Thus, we getcsc⁡−1(−1)=−π2{\csc ^{ - 1}}\left( { - 1} \right) = - \dfrac{\pi }{2}

Therefore, the exact value of csc⁡−1(−1)=−π2{\csc ^{ - 1}}\left( { - 1} \right) = - \dfrac{\pi }{2}.

Note:
We know that Trigonometric Equation is defined as an equation involving trigonometric ratios. Trigonometric identity is an equation that is always true for all the variables. We should know that we have many trigonometric identities that are related to all the other trigonometric equations. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angle. Trigonometric Ratios are used to find the relationships between the sides of a right-angle triangle. We should remember that all the trigonometric ratios can be written in the form of a basic Trigonometric Ratio of sine and cosine.