What is the Value of Cosec 45 Degrees ?

Solution :

The value of cosec 45 degrees is \(\sqrt{2}\).

Proof :

Let ABC be a triangle, right angled at B, in which \(\angle\) A = \(\angle\) C = 45 degrees45 degrees

\(\therefore\)  BC = AB

Let  AB = BC = a

Then by pythagoras theorem,

\(AC^2\) = \(AB^2\) + \(BC^2\) = \(a^2\) + \(a^2\) = \(2a^2\)

\(\implies\)  AC = \(\sqrt{2}a\)

In \(\Delta\) ABC,  \(\angle\) C = 45 degrees

By using trigonometric formulas,

\(cosec 45^{\circ}\) = \(hypotenuse\over perpendicular\) = \(h\over p\)

\(cosec 45^{\circ}\)  = hypotenuse/side opposite to 45 degrees = \(AC\over AB\) = \(\sqrt{2}a\over a\) = \(\sqrt{2}\)

Hence, the value of \(cosec 45^{\circ}\) = \(\sqrt{2}\)

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