Question

How are the six basic trigonometric functions related to right triangles?

Answer 1

Hint: In the given question, we have been asked how are the basic trigonometric functions and right triangles related. To answer this, we need to have a firm knowledge of the basics. First, we are going to draw a right triangle, then express the basic trigonometric functions with respect to the sides of the triangle.

Complete step-by-step answer:
The six basic trigonometric functions are:
sine, \[\sin \theta \]
cosine, \[\cos \theta \]
secant, \[\sec \theta \]
cosecant, \[{\mathop{\rm cosec}\nolimits} \theta \]
tangent, \[\tan \theta \]
cotangent, \[\cot \theta \]

Let there be a right triangle, \[\Delta ABC\], right-angled at \[C\]. Let the reference angle be \[\angle A\] and be represented by \[\theta \]. The longest side of the triangle is the “hypotenuse”, the side next to the angle is the “adjacent” and the side opposite to it is the “opposite”.
\[\sin \theta = \dfrac{{opp}}{{hyp}} = \dfrac{a}{h}\]
\[\cos \theta = \dfrac{{adj}}{{hyp}} = \dfrac{b}{h}\]
\[\sec \theta = \dfrac{{hyp}}{{adj}} = \dfrac{h}{b}\]
\[{\mathop{\rm cosec}\nolimits} \theta = \dfrac{{adj}}{{hyp}} = \dfrac{h}{a}\]
\[\tan \theta = \dfrac{{opp}}{{adj}} = \dfrac{a}{b}\]
\[\cot \theta = \dfrac{{adj}}{{opp}} = \dfrac{b}{a}\]

Note: In the given question, we were asked to show how a right triangle and the six trigonometric functions are related. To do that, we first drew a right triangle, then we wrote the formulae of the trigonometric functions expressed by hypotenuse, perpendicular and base. Then we expressed them with respect to the sides of the triangle. So, it is really important that we know the formulae of all the trigonometric functions and where, when and how to use them so that we can get the correct result.