Trigonometry concepts

In this topic, we will learn the basic concepts of trigonometry and trigonometric ratios.

The name “trigonometry” is made up of two Greek words “trignon” which means “triangle” and “metron” which means measure.  Trigonometry is the branch of mathematics that deals with the relationship between the length of sides and angles of triangle. 

In its geometrical applications we mainly use trigonometry in right angled triangles.

Let ΔABC be a right angled triangle right angled at B.

So the trigonometric ratios of the triangle w.r.t ∠A can be written as.

sine A =\frac { side\quad opposite\quad of\quad \angle A\quad or\quad perpendicular(p) }{ hypotenuse(h) } = \frac { BC }{ AC }
 
cosine A =\frac { side\quad adjecent\quad of\quad \angle A\quad or\quad base(b) }{ hypotenuse(h) } =\frac { AB }{ AC }
 
tangent A =\frac { side\quad opposite\quad of\angle A\quad or\quad perpendicular(p) }{ side\quad adjecent\quad of\quad \angle A\quad or\quad base\angle (b) } =\frac { BC }{ AB }
 
cosecant A =\frac { hypotenuse(h) }{ side\quad opposite\quad of\quad \angle A\quad or\quad perpendicular(p) } =\frac { AC }{ BC }
 
secant A =\frac { hypotenuse(h) }{ side\quad adjecent\quad of\quad \angle A\quad or\quad base(b) } =\frac { AC }{ AB }
 
cotangent A =\frac { side\quad adjecent\quad of\quad \angle A\quad or\quad base(b) }{ side\quad opposite\quad of\quad \angle A\quad or\quad perpendicular(p) } =\frac { AB }{ BC }
 

In general we can say if θ is an angle in a right angle triangle, then the trigonometric ratios will be:

sin θ=\frac { p }{ h }
cosec θ=\frac { h }{ p }
cosθ =\frac { b }{ h }
sec θ=\frac { h }{ b }
tan θ=\frac { p }{ b }
cot θ=\frac { b }{ p }

Note following are the relationship between the trigonometric rations:

1) cosec θ =\frac { 1 }{ sin\theta }
2) sec θ=\frac { 1 }{ cos\theta }
3) cot θ=\frac { 1 }{ tan\theta }
4) tan θ =\frac { sin\theta }{ cos\theta }

 

Let us see this with an example,

Question- Let ΔABC be a right angled triangle, right angled at B. Where AB = 4 cm , BC = 3 cm and AC = 5 cm. find the trigonometric ratios w.r.t to ∠A.

Sol- In the given triangle ΔABC,

AB= 4 cm

BC = 3 cm

AC= 5 cm

So , while taking the trigonometric ratios w.r.t. angle ∠A,

The perpendicular (p) will be the side opposite of ∠A = BC

And the base(b) will be the side adjacent to∠ A = AB.

So, sin A = \frac { perpendicular(p) }{ hypotenuse(h) } =\frac { 3 }{ 5 }

      cos A = \frac { base(b) }{ hypotenuse(h) }    = \frac { AB }{ AC } = \frac { 4 }{ 5 }

       tan A = \frac { perpendicular(p) }{ base(b) } = \frac { AB }{ BC } = \frac { 3 }{ 4 }

       cosec A = \frac { 1 }{ sinA } = \frac { AC }{ BC } = \frac { 5 }{ 3 }

         sec A = \frac { 1 }{ cosA } =\frac { AC }{ AB } = \frac { 5 }{ 4 }  

          cot A = \frac { 1 }{ tanA } = \frac { BC }{ AB } = \frac { 4 }{ 3 }


Now that you have understood the basics of trigonometry let us look at the trigonometric table.

TRIGONOMETRIC TABLE

Please click the above link for learning the trigonometric table.