Express \(Sine(sin)\), \(Cosine(cos)\) and \(tangent(tan)\) in terms of sides of triangle

Question: Express \(Sine(sin)\), \(Cosine(cos)\) and \(tangent(tan)\) in terms of sides of triangle 

Answer:

For a right angle triangle  we can find sine or sin, cosine or cos, tangent or tan in terms of their side. 

To know more accurately about sin, cos and tan we must have to know about (1)base, (2)height (perpendicular) and (3)hypotenuse.

Figure-1 (right angle triangle)

   In general we use base, hypotenuse and height(perpendicular) as shown in figure.  But this method is not always right.   Let's know what is right

(1) Definition of hypotenuse:     

The side opposite to the right angle or \(90^{o}\) is called hypotenuse, it is the largest side of right angle triangle. 

(2) Definition of perpendicular: 

It depends on the angle which we are going to consider. If we consider angle C the perpendicular is AB. 

If we consider angle A then perpendicular is  BC. (See below figure-2) 

so height is the opposite side of the angle which we have consider. 

(3) Definition of base:

Then the rest side is called base. 

Figure-2

Now,   

One can say that [considering angle \(\angle C\) ]

(1) Sine is the ratio of perpendicular and hypotenuse

\[sin C=\frac{perpendicular(AB)}{hypotenuse(AC)}\]

(2) cos is the ratio of base and hypotenuse

\[cos C=\frac{base(BC)}{hypotenuse(AC)}\]

(3) tan is the ratio of base and perpendicular

\[tan C=\frac{perpendicular(AB)}{base(BC)}\]

Again,

One can say that [considering angle \(\angle A\) ]

(1) Sine is the ratio of perpendicular and hypotenuse

\[sin C=\frac{perpendicular(BC)}{hypotenuse(AC)}\]

(2) cos is the ratio of base and hypotenuse

\[cos C=\frac{base(AB)}{hypotenuse(AC)}\]

(3) tan is the ratio of base and perpendicular

\[tan C=\frac{perpendicular(BC)}{base(AB)}\]

So, in this way one can easily get \(sin, cos, tan\) ratio. And using this one can find \(cosec, sec, cot\). 

[As \(cosec\theta =\frac{1}{sin\theta }\) ; \(sec\theta =\frac{1}{cos\theta }\) and \(cot\theta =\frac{1}{tan\theta }\) ]