Question

How do you find the value of \[\csc (\dfrac{\pi }{2})\]?

Answer 1

Hint: To find this just use the basic trigonometric relations. As we have to find the value of \[\csc (\dfrac{\pi }{2})\] and we know that \[\csc \theta \] is reciprocal of \[\sin \theta \]and we know the value of \[\sin \theta \]. Here \[\theta \] is \[\dfrac{\pi }{2}\] and we have known the value of \[\sin \left( \dfrac{\pi }{2} \right)\] is \[1\] and the reciprocal of \[1\] is also \[1\]. So we have calculated the value of \[\csc (\dfrac{\pi }{2})\] and that is \[1\].

Complete step by step solution:

Let’s assume a right-angled triangle with right angle \[\angle C\].
Let \[\angle A\] be \[\theta \]
Since we know that,
\[\sin \theta =\dfrac{base}{hypotenuse}\]
Also, we have a basic trigonometric relation
\[\cos ec\theta =\dfrac{1}{\sin \theta }\]
And we also know that \[\sin \dfrac{\pi }{2}=1\]
\[\Rightarrow \csc \left( \dfrac{\pi }{2} \right)=\dfrac{1}{\sin \left( \dfrac{\pi }{2} \right)}\]
Now substitute value of \[\sin \dfrac{\pi }{2}=1\]
\[\Rightarrow \csc \left( \dfrac{\pi }{2} \right)=\dfrac{1}{1}=1\]

Thus, we have calculated the value of \[\csc \left( \dfrac{\pi }{2} \right)=1\]
Hence,\[\csc \left( \dfrac{\pi }{2} \right)=1\]

Note:
When we have to calculate these basic trigonometric values just use the basic trigonometric relations. Also remember that side opposite to angle \[\theta \] is perpendicular and opposite to right angle is hypotenuse, now according to figure above if hypotenuse is \[1\] and angle is \[\dfrac{\pi }{2}\] then opposite side perpendicular is also \[1\].