Trigonometric Functions of Special Angles

Trigonometry  is branch of  mathematics which deals with triangles.Trigonometry means measure of triangles . Trigonometry deals with the relationship between sides and angles of triangle. There are six basic trigonometric functions and they are sine, cosine, tangent, secant, cosecant and cotangent. They are defined based on the sides of a right triangle.

Explanation of Trigonometric Functions of Special Angles

Basic functions of  a right triangle following are defined as follows:



i)Sine abbrevated as "sin"                and   sin(theta) = ` c/b`

ii)Cosine abbrevated as "cos"          and   cos(theta) = ` a/b`

iii)Tangent abbrevated as "Tan"         and   Tan(theta) = ` c/a`

iv)Secant abbrevated as  "sec"         and   sec(theta) = ` b/a`

v)Cosecant abbrevated as "cosec"    and   cosec(theta) = ` b/c`

vi)cotangent abbrevated as "cot"       and   cot(theta) = ` a/c`

There are 4 types trigonometric identities.

1:  Reciprocal Identiy :

i) cosecA = `(1)/(sin A)`

ii) sec A= `(1)/(cos A)`

iii) cotA = `(1)/(tanA)`

2:  Pythagorean Identites :

Sin2 A  + Cos2 `A =1`

Sec2 `A`   - Tan2 `A = 1<br>`

Cosec2 A  - Cot2 A`=1`

3:  Quotent Identites :-

tan`A` =  sin`A`` /` cos`A`

cotA =  cos`A`` /` sin`A`

4 : Even- odd Identites:

1)sin(-`A) =`sin`A`

2)cos(-`A) = cosA`

`3)`tan(-`A) =`tan`A`

`4)`cosec(-`A)= `-cosec`A`

`5)`sec(-`A) = `sec`A`

`6)cot(-A) = -cot(A)<br>`

Chart for trigonometric functions of special angles :-

here "U" stands for undefined or infinite.

Examples on Trigonometric Functions of Special Angles:

Given an right  triangle whose base is 3 cms and base angle with hypotenuse is 60o.find the following

i) height of triangle

ii) area of triangle

Solution:

Given an right  triangle and base side is 3 cms and base angle is 60o.

Tan of base angle is height /base

so tan 60o = height /3

=> `sqrt(3)`  = height /3

=>height of triangle is 3`sqrt(3)` cms.

ii)

Area of a right triangle A is 1/2(base * height)

=> A =(1/2)*3*3`sqrt(3)`

`=> A=4.5``sqrt(3)`

`A= 7.79`


Ex 2 :   In a right triangle given angle A is 60o and side adjacent to angle A is 5cms .

find the hypotenuse of triangle

solution:

Cos A = adjacent side/hypotenuse

Cos 60o = 5/hypotenuse

=>1/2=5/hypotenuse

=> hypotenuse =5/2 =

=> hypotenuse side= 2.5cms

Ex3:

find value of cos 60o + sin 30o

solution:

value of cos 60o=1/2

value of sin 30o=1/2

hence

cos 60o + sin 30o  =1/2 +1/2 =1