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TRIGNOMETRY
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Trigonometry = tri+ gon + metry
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)
Study of relationship between sides and angles of a triangle
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1
...
BC is base
(side opposite to angle A), AB is perpendicular (side opposite to angle C) and AC is hypotenuse
A
(side opposite to angle B)

Perpendicular

90
°°

B

Base

C

Fig
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To find the trigonometric ratios in aright angled triangle –
According to Pythagoras theorem,
(Hypotenuse) 2 = (Perpendicular) 2 + (Base) 2
i
...
,

(AC) 2 = (AB)2 + (BC)2

In the same, way SinΘ = 𝐴𝐵/𝐴𝐶 = 𝑃/𝐻,

CosΘ = 𝐵𝐶/𝐴𝐶 = 𝐵/𝐻, tanΘ = 𝐴𝐵/𝐵𝐶 = 𝑃/𝐵

CosecΘ = 1/ 𝑠𝑖𝑛 𝛩 = 𝐻/𝑃,

secΘ = 1 /𝑐𝑜𝑠 𝛩 = 𝐻/𝐵,

cot Θ = 1/𝑡𝑎𝑛 𝛩 = 𝐵/𝑃

Therefore, by applying these formula we can calculate value of sin , cos ,tan ,cosec, sec and cot
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To ease our process of
learing we could easily use the desi nuska i
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Question: In a triangle ABC, right angled at B, AB = 8cm and BC =6cm find
A

a) SinA and CosA
b) SinC and CosC
B

C

Answer- a) SinA = 3/5, CosA = 4/5
b)SinC = 4/5
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Trigonometric Ratios of Some Specific Angles

From the table it is noted that angle A increases from 0° to 90°, sinA increases from 0 to 1 and
CosA decreases from 1 to 0
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g
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2
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Hence the
derivation is just for learning, the corresponding values had to be learned which has been note
down in the tabular form above
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Determine the lengths of the sides BC and AC?
Answer: BC = 5√3 cm and AC = 10 cm

Trigonometric Ratios of Complementary Angles
We know if ,sinB =

𝐴𝐶
𝐵𝐶

, COS B =

𝐵𝐴
𝐵𝐶

, TanB =

𝑆𝑖𝑛 𝐵
𝐶𝑜𝑠𝐵

=

𝐴𝐶
𝐵𝐶

,

CosecB =

1

𝐵𝐶

=
𝐵

𝑆𝑖𝑛

, secB =

𝐴𝐶

1

=
𝐶𝑜𝑠𝑒𝑐

𝐵𝐶
𝐴𝐵

, CotB =

1

=
𝑇𝑎𝑛

𝐵𝐶
𝐴𝐶

(1)

Now; AB is the side opposite & AC is the side adjacent to the angle
Sin( 900 – B)=

𝐴𝐵

Cos (900- B)=

𝐴𝐶

,

𝐵𝐶

𝐵𝐶

Tan (900-B) =

𝐵𝐴
𝐴𝐶
𝐵𝐶

Cosec (900 – B) =
Sec (900 – B) =
Cot (900 – B) =

𝐵𝐴

𝐵𝐶
𝐴𝐶
𝐴𝐶
𝐵𝐴

Now compare the ratio in (1) & ( 2)
Sin (900 – B) =

𝐵𝐴

Cos (900 – B) =

𝐴𝐶

tan (900 – B) =

= Cos B

𝐵𝐶

𝐵𝐶
𝐵𝐶
𝐴𝐶

= Cot B

Cosec (900 – B) =
Sec (900 – B) =
Cot (900 – B) =

𝐵𝐶
𝐴𝐶
𝐴𝐶
𝐵𝐴

= SinB

𝐵𝐶
𝐵𝐴

= Sec B

= CosecB
= tanB

Summarizing the above things we get from the following conclusions :Sin (900 – B) = Cos B, Cos (900 – B) = SinB
tan (900 – B) = Cot B, Cot (900 – B) = tanB
Cosec (900 – B) = Sec B, Sec (900 – B) = CosecB
Value of angle B may be 0ͥͦͦͦͦ0 or 900

Note




Tan00 = Cot 900
Sec 00=cosec 900 = sec 900 = 1
Cosec 00= Tan 9 00 & Cot00 are not defined

Question:Prove that

𝒄𝒐𝒕𝑨−𝐜𝐨𝐬 𝑨

𝒄𝒐𝒔𝒆𝒄𝑨−𝟏

𝐜𝐨𝐭

𝒄𝒐𝒔𝒆𝒄𝑨+𝟏

=
𝑨+𝒄𝒐𝒔𝑨

Trigonometric Identities
There are mainly three identities used for solving question
1) Sin2B+Cos2B=1
2) 1 + Tan2B =Sec2B
3) Cot2B + 1 =Cosec2B
Question:1
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(secA + tanA) (1 – sinA)
Answers- 1
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cosA



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