Question

What is \[\theta \] equal to and why? \[\csc \theta =1\].

Answer 1

Hint: In this problem, we have to find the value of \[\theta \], where \[\csc \theta =1\]. We should first know that if \[\theta \] is one of the acute angles in a triangle, then the sine of theta is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse and the tangent is the ratio of the opposite side to the adjacent side.

Complete step-by-step solution:
Here we have to find the value of \[\theta \] form.
We know that the given trigonometric value is,
\[\Rightarrow \csc \theta =1\]……. (1)
 We know that, \[\csc \theta =\dfrac{1}{\sin \theta }\].
We can now write (1) as,
\[\Rightarrow \dfrac{1}{\sin \theta }=1\]
We can now multiply \[\sin \theta \] on both the left-hand side and the right-hand side in the above step, we get
\[\Rightarrow 1=\sin \theta \]
We can now multiply the arc sin or the sine inverse function on both the left-hand side and the right-hand side in the above step, we get
\[\Rightarrow {{\sin }^{-1}}\sin \theta ={{\sin }^{-1}}1\]
We can now cancel the sine and the sine inverse in the above step, we get
\[\Rightarrow \theta ={{\sin }^{-1}}1\]
We know that the value of sine will be 1, if \[\theta \] is equal to \[\dfrac{\pi }{2}+2n\pi \], where n is the integer.
Therefore, the value of \[\theta =\dfrac{\pi }{2}+2n\pi \].

Note: Students should know that, if \[\theta \] is one of the acute angles in a triangle, then the sine of theta is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse and the tangent is the ratio of the opposite side to the adjacent side. We should also know that we can cancel the arc sine and the sine function as they are inverse to each other.