The trigonometric function gives the ratio of different sides of a right-angle triangle.The correct option is D, arc cosec (x) = arc sin (1/x).
What are Trigonometric functions?
The trigonometric function gives the ratio of different sides of a right-angle triangle.
[tex]\rm Sin \theta=\dfrac{Perpendicular}{Hypotenuse}\\\\\\Cos \theta=\dfrac{Base}{Hypotenuse}\\\\\\Tan \theta=\dfrac{Perpendicular}{Base}\\\\\\Cosec \theta=\dfrac{Hypotenuse}{Perpendicular}\\\\\\Sec \theta=\dfrac{Hypotenuse}{Base}\\\\\\Cot \theta=\dfrac{Base}{Perpendicular}\\\\\\[/tex]
where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.
Assume a right-angled triangle such that the hypotenuse of the triangle is of 1 unit, while the side opposite to angle A measures 1/x units, as shown in the triangle below. Also, the measure of the ∠A is θ. Therefore, we can write,
sin θ = 1/x
cosec θ = x
Solving both the ratios for θ, we will get,
θ = arc sin (1/x)
θ = arc cosec (x)
Equating the two equations formed above together, therefore, we can write,
θ = θ
arc cosec (x) = arc sin (1/x)
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