Trigonometry

Trigonometry (trigonon "triangle" + metron "measure") is a branch of mathematics that studies triangles, particularly right triangles. Trigonometry deals with relationships between the sides and the angles of triangles, and with trigonometric function, which describe those relationships and angles in general, and the motion of waves such as sound and light waves.

Read the following sentence: 

Some Of Her Children Are Having Trouble  Over Algebra

Break the sentence in 3 parts (different parts r shown wid different colour)

Some Of Her  Children Are Having  Trouble Over Algebra

U have to consider only 1st letter of each word.

1st letter of 1st word of each part(S,C,T) represent name of Angle.

Means- S-sine, C-cosine, T-tangent

And 1st letter of the remaining two words of each part(Of Her,Are Having,Over Algebra) represent the name of sides. Means,

O-opposite side , H-hypotenuse , A-adjacent side

 In right angle ΔABC, ∟A  &∟C are acute ∟s.

Out of these 2 acute ∟s v will consider ∟A.

Since v have chosen ∟A, so side opposite to ∟A is CB , adjacent side to ∟A is AB, & hypotenuse is AC .

NOTE: If v consider ∟C in place of  ∟A, the position of sides will change. In that case ,side opposite to  ∟C  will be AB and side adjacent to  ∟C will be CB ,hypotenuse will be AC .In right angle triangle, hypotenuse is a fixed side, like in the given  ΔABC side AC is hypotenuse. Opposite side and adjacent side will depend upon which acute ∟ v are considering.

Thus(trigonometric ratios of ∟A in right ∟tri)

SineA = Opposite ÷ Hypotenuse (where A is any 1 acute ∟of right ∟ tri)

CosineA = Adjacent ÷ Hypotenuse

TangentA = Opposite ÷ Adjacent

Angles of Sine

1st step:

Write nos.frm 0 to 4

1)0

2)1

3)2

4)3

5)4

2nd step:

Divide each no. by 4(as 4 is largest of all)

1) 0/4=0

2) ¼

3) 2/4=1/2

4) ¾

5) 4/4=1

3rd step:

Take square root of the resultant

1) √0=0

2) √1/4=1/2

3) √1/2=1/√2

4) √3/4=√3/2

5) √1=1

4th step:

The values in last step shows the values of sin0°,sin30°,sin45°,sin60°,sin90°

So we have

Sin0°=0

Sin30°=1/2

Sin45°=1/√2

Sin60°=√3/2(its root 3 upon 2, root is not over 2)

Sin90°=1

Note : As the angle increases, value of angle is also increasing.

Angles of Cosine

U don’t actually have to learn the values of Cosine and Tangent, if u have learnt the values of Sine. What u have to do is to write the values of Sine angles in reverse order for getting values of Cosine Angle. This means

Cos0°=1

Cos30°= √3/2(its root 3 upon 2,root is not over 2)

Cos45°=1/√2

Cos60°=1/2

Cos90°=0

Note: Sin 45°= Cos 45°
Angles of Tangent:

U have to use formula SinA/Cos A=Tan A where A is any acute angle.

Sin0°/cos0°=tan0°

Put value of these 2,i.e 0/1=0,so tan0°=0.

Similarly tan30°=1/√3

Tan45°=1

Tan60°=√3

Tan90°=∞ (b’coz 1/0=∞ i.e not defined)

Take reciprocal of SinA to get CosecA:

Cosec 0°=1/0=∞

Cosec 30°=2/1=2

And so on

Similarly take reciprocal of CosA which is SecA,n reciprocal of TanA is CotA.

Trigonometric identities-

An equation involving trigonometric ratios of an ∟is called trigonometric identity,if it is true 4 all values of the ∟(s) involved.

Basically there r 9 identities, I’l tell u tricks 2 learn all these.

U have 2 learn only 1 basic identity that is

1. sin²A + cos²A = 1 ---------------(equ 1) (for 0°≤A≤90° read it as where A can b greater thn or equal to 0° n less or equal to 90°)

frm (equ 1) u can get 2 more iden.

2. sin²A = 1- cos²A

3. cos²A = 1 - sin²A

Now what u have 2 do is to divide (equ 1) by sin²A

sin²A/sin²A + cos²A/sin²A = 1/sin²A

u’I get another iden.

4. 1 + cot²A = cosec²A

(where A can b  greater thn 0 bt less thn or equal to 90)
Frm this u can get 2 more iden.

5. cot²A = cosec²A – 1

6. cosec²A – cot²A = 1

Again take (equ 1) n now divide it by cos²A
sin²A/co²sA + cos²A/cos²A = 1/cos²A

So,here is new iden.

7. tan²A + 1 = sec²A
(where A can b greater thn or equal to 0° n less thn 90°)

Frm this u can get last 2 iden.

8. tan²A = sec²A – 1

9. sec²A – tan²A = 1