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How do you evaluate $\csc 180$ ?

Last updated date: 17th Jul 2024
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Hint: In this question, we have to find the value of the cosecant of 180 degrees, cosecant is a trigonometric function. So to solve the question, we must know the details of trigonometric functions and how to find their values. The trigonometry helps us to find the relation between the sides of a right-angled triangle that is the base, the perpendicular and the hypotenuse. The trigonometric functions have different signs in different quadrants, so by using the above information, we can find out the cosecant of the given angle.

Complete step by step solution:
We know that –
  \csc x = \dfrac{1}{{\sin x}} \\
   \Rightarrow \csc 180 = \dfrac{1}{{\sin 180}} \\
$\sin (180^\circ ) = \sin (180^\circ + 0^\circ )$
We know that sine is negative in the third quadrant, so –
  \sin (180^\circ ) = - \sin (0^\circ ) \\
   \Rightarrow \sin (180^\circ ) = 0^\circ \\
   \Rightarrow \csc 180 = \dfrac{1}{0} = \infty \\
Hence, the value of $\csc (180^\circ )$ is not defined.

Note: Sine, cosine and tangent are the main functions of the trigonometry while cosecant, secant and cotangent functions are their reciprocals respectively. All the trigonometric functions have a positive value in the first quadrant. In the second quadrant, sine is positive while all the other functions are negative; in the third quadrant, tan function is positive while all other functions are positive; and in the fourth quadrant, cosine function is positive while all others are negative, that’s why $\sin (180 + \theta ) = - \sin \theta $ . We also know that the trigonometric functions are periodic, we know the value of the cosecant function when the angle lies between 0 and $\dfrac{\pi }{2}$ . That’s why we use the periodic property of these functions to find the value of the cosecant of the angles greater than $\dfrac{\pi }{2}$ .