Question

Which of the following pairs is a correct trigonometric inter-relationship?

1. \[\cos \theta \]	a. \[\dfrac{{\cos \theta }}{{\sin \theta }}\]
2. \[\tan \theta \]	b. \[\dfrac{1}{{\text cosec \theta }}\]
3. \[\cot \theta \]	c. \[\dfrac{1}{{\sec \theta }}\]
4. \[\sin \theta \]	d. \[\dfrac{1}{{\cot \theta }}\]
	e. \[\sin \theta \cos \theta \]

A. \[1 - d,2 - e,3 - b,4 - a\]
B. \[1 - b,2 - a,3 - e,4 - d\]
C. \[1 - c,2 - d,3 - a,4 - b\]
D. \[1 - e,2 - b,3 - c,4 - d\]

Answer 1

Hint: In this question we have to find the correct pair of trigonometric inter- relationship. First, we will consider a right-angle triangle. Concerning any of the acute angles, we will find the relationship between the sides and the angle. From there we will find the trigonometric inter-relationship. Finally we can find which is the correct option.

Complete step-by-step answer:
Here, some trigonometric relationships are given. We have to find the correct trigonometric inter-relationship.
Let us consider a right-angle triangle.
               

Here, \[\Delta ABC\] is a right-angle triangle whose \[\angle B = {90^ \circ }\] and other two angles are acute angles.
Let us take \[\angle C = \theta \].
We know that the opposite side of an acute angle of a right-angle triangle is perpendicular, the opposite side of the right-angle is hypotenuse and the remaining side is known as base.
With respect to the \[\angle C = \theta \], \[AB\] is the perpendicular, \[BC\] is the base and \[AC\] is the hypotenuse.
Now, we will find the trigonometric relationship.
Form the relation between sides and angles of a right-angle triangle we get,
\[
  \sin \theta = \dfrac{1}{{\text cosec \theta }} \\
  \cos \theta = \dfrac{1}{{\sec \theta }} \\
  \tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }} = \dfrac{1}{{\cot \theta }} \\
  \cot \theta = \dfrac{{\cos \theta }}{{\sin \theta }} \\
 \]
Form the given relationships we get,
\[\dfrac{{\cos \theta }}{{\sin \theta }} = \cot \theta \]
\[\dfrac{1}{{\text cosec \theta }} = \sin \theta \]
\[\dfrac{1}{{\sec \theta }} = \cos \theta \]
\[\dfrac{1}{{\cot \theta }} = \tan \theta \]

So, the correct relations are: \[1 - c,2 - d,3 - a,4 - b\]

So, the correct answer is “Option C”.

Note: Let us consider \[\Delta ABC\] is a right-angle triangle whose \[\angle B = {90^ \circ }\] and other two angles are acute angles.
Let us take \[\angle C = \theta \].
Form the relation between sides and angles of a right-angle triangle we get,
\[\sin \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Hypotenuse}}}}\]
\[\cos \theta = \dfrac{{{\text{Base}}}}{{{\text{Hypotenuse}}}}\]
\[\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}\]
\[\text cosec \theta = \dfrac{{{\text{Hypotenuse}}}}{{{\text{Perpendicular}}}}\]
\[\cos \theta = \dfrac{{{\text{Hypotenuse}}}}{{{\text{Base}}}}\]
\[\cot \theta = \dfrac{{{\text{Base}}}}{{{\text{Perpendicular}}}}\]
The angle which is greater than \[{0^ \circ }\] but less than \[{90^ \circ }\], is called the acute angle.