Trigonometry

Formulas
FSC (Part-I)
(Pre- Engineering, ICS)

Prepared by:
Prof. Ali Raza Khan
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 1 of 11
CH # 09
Fundamentals of Trigonometry
Conversion from Radian to Do M′S″
and Do M′S″ to Radian Note:
i.
𝜋
1𝑜 = 180 𝑟𝑎𝑑 Do M′S″
(Sexadecimal system)
ii. 1 = 60′ (Minute)
𝑜 Stand for Degree, Minute and second.
Note:
iii. 1′ = 60″ (Seconds)
Do = Degree
Radian to Do M′S″ M′ =Minute
180𝑜
iv. 1 𝑟𝑎𝑑 = S″ =Second
𝜋
′ 1 𝑜 Note:
v. 1 = (60) 1o = 0.0175 rad
1 1 rad = (57.296)o
vi. 1″ = (3600)𝑜
Radian:
Radian is the measure of an angle subtended at the center of the circle by an Arc, whose
length is equal to the radius of the circle.

Relation between ‘l’ and ′𝜃’:

i. 𝑙 = 𝑟𝜃
𝑙
ii. 𝑟=𝜃
𝑙
iii. 𝜃=𝑟
Trigonometry Ratios Functions:
𝑃𝑒𝑟𝑝 𝐵𝑎𝑠𝑒 𝑃𝑒𝑟𝑝
• 𝑠𝑖𝑛𝜃 = 𝑐𝑜𝑠𝜃 = 𝑡𝑎𝑛𝜃 =
Hyp Hyp Base

1 1 1
• 𝑠𝑖𝑛𝜃 = (𝑐𝑜𝑠𝑒𝑐𝜃)
𝑐𝑜𝑠𝜃 = 𝑠𝑒𝑐𝜃
𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑡𝜃
1 1 1
• 𝑐𝑜𝑠𝑒𝑐𝜃 = 𝑠𝑒𝑐𝜃 = 𝑐𝑜𝑡𝜃 =
𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝜃
𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃
• 𝑐𝑜𝑡𝜃 = 𝑡𝑎𝑛𝜃 =
𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃

How To Remember Note: (Stands for)

Some People Have 𝑆𝑖𝑛𝜃 = 𝑃⁄𝐻 Perp = Perpendicular
Hyp = Hypotenuse
Curly Brown Hair 𝐶𝑜𝑠𝜃 = 𝐵⁄𝐻 Note:
Through Proper Brush 𝑇𝑎𝑛𝜃 = 𝑃⁄𝐵 𝒄𝒐𝒔𝒆𝒄𝜽 𝑎𝑙𝑠𝑜 𝑤𝑟𝑖𝑡𝑒𝑛 𝑎𝑠 𝒄𝒔𝒄𝜽

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 2 of 11
FundamentalIdentities:
1. 𝑠𝑖𝑛 2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 2. 1 + 𝑡𝑎𝑛 2𝜃 = 𝑠𝑒𝑐 2 𝜃
i. 𝑠𝑖𝑛 2 𝜃 = 1 − 𝑐𝑜𝑠 2 𝜃 i. 1 = 𝑠𝑒𝑐 2 𝜃 − 𝑡𝑎𝑛 2 𝜃
ii. 𝑐𝑜𝑠 2 𝜃 = 1 − 𝑠𝑖𝑛2 𝜃 ii. 𝑡𝑎𝑛 2 𝜃 = 𝑠𝑒𝑐 2 𝜃 − 1
3. 1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑜𝑠𝑒𝑐 2 𝜃
i. 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑜𝑠𝑒𝑐 2 𝜃 − 1
ii. 1 = 𝑐𝑜𝑠𝑒𝑐 2 𝜃 − 𝑐𝑜𝑡 2 𝜃

Quadrants:
How to Remember:
I II III Iv
Add Sugar To Coffee
All Sin 𝜃 Tan 𝜃 Cos 𝜃
cosec 𝜃 Cot 𝜃 Sec 𝜃

Note:
Sin(−𝜃) = − Sin 𝜃 Cosec(−𝜃) = − Cosec 𝜃
Cos(−𝜃) = Cos 𝜃 Sec(−𝜃) = Sec 𝜃
Tan(−𝜃) = − Tan 𝜃 Cot(−𝜃) = − Cot 𝜃
Note:
𝜋 180𝑜
𝜋 = 180𝑜 = = 60𝑜
3 3
𝜋 180𝑜
2𝜋 = 360𝑜 = = 45𝑜
4 4
𝜋 180𝑜 𝜋 180𝑜
= = 90𝑜 = = 30𝑜
2 2 6 6
3𝜋 3 × 180𝑜 2𝜋 2 × 180𝑜
= = 270𝑜 = = 120𝑜
2 2 3 3

• 1 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝑨𝒏𝒕𝒊 − 𝒄𝒍𝒐𝒄𝒌𝒘𝒊𝒔𝒆) = 360𝑜

1
• 2 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝐴𝑛𝑡𝑖 − 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒) = 180𝑜
1
• 𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛(𝐴𝑛𝑡𝑖 − 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒) = 90𝑜
4

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 3 of 11

Table
Degree 0𝑜 30𝑜 45𝑜 60𝑜 90𝑜 180𝑜 270𝑜 360𝑜
𝝅 𝝅 𝝅 𝝅 𝟑𝝅
Radian 0 𝝅 2𝝅
𝟔 𝟒 𝟑 𝟐 𝟐
1 1 √3
Sin 𝜃 0 2 √2 2 1 0 −1 0
0.5 (0.707) (0.8660)
√3 1 1
Cos 𝜃 1 2 √2 2 0 −1 0 1
0.8660) (0.707) 0.5
1
√3 ∞ −∞
Tan 𝜃 0 √3 1 0 0
(0.577) (1.732) 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑈𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
−1
Note: 𝑐𝑜𝑠1200 = 𝑡𝑎𝑛1350 = −1
2

CH # 10
Trigonometry Identities
Note:
If 𝜃 is added to or subtracted from odd multiple of right angle , Trigonometric ratios change in-to
co-ratios and vice versa.
Nota:
Example:
𝑆𝑖𝑛 ⇌ 𝐶𝑜𝑠
𝜋 3𝜋
Sin ( 2 − 𝜃) = 𝑐𝑜𝑠 Cos ( 2 − 𝜃) = 𝑆𝑖𝑛𝜃
𝑆𝑒𝑐 ⇌ 𝐶𝑜𝑠𝑒𝑐
𝜋 3𝜋 𝜋 𝑇𝑎𝑛 ⇌ 𝐶𝑜𝑡
𝑎𝑛𝑑 𝑎𝑟𝑒 𝑜𝑑𝑑 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓
2 2 2
Note:
Trigonometric Ratios are changed on 𝟗𝟎𝒐 and 𝟐𝟕𝟎𝒐. These two angles come on the y-axis. And
Trigonometric Ratios these are not changed on 𝟏𝟖𝟎𝒐 and 𝟑𝟔𝟎𝒐. ( X-axis )

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 4 of 11
Fundamental Law of Trigonometry:
Measure of the angle Quadrant
• cos(𝛼 − 𝛽 ) = 𝑐𝑜𝑠𝛼 𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛽 𝜋
− 𝜃 I
• cos(𝛼 + 𝛽 ) = 𝑐𝑜𝑠𝛼 𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛽 𝜋
2
+ 𝜃 𝑜𝑟 𝜋 − 𝜃 II
• sin(𝛼 + 𝛽 ) = 𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛽 2
3𝜋
• sin(𝛼 − 𝛽 ) = 𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛽 2
− 𝜃 𝑜𝑟 𝜋 + 𝜃 III
tan𝛼+tan𝛽 3𝜋
• tan(𝛼 + 𝛽 ) = 1−tan𝛼tan𝛽 2
+ 𝜃 𝑜𝑟 2𝜋 − 𝜃 IV

tan𝛼−tan𝛽
• tan(𝛼 − 𝛽 ) =
1+tan𝛼tan𝛽
Deductions from Fundamental Law:
𝜋 𝜋 𝜋
sin ( − 𝜃) = 𝑐𝑜𝑠𝜃 cos ( − 𝜃) = 𝑠𝑖𝑛𝜃 tan ( − 𝜃) = 𝑐𝑜𝑡𝜃
{ 2 2 2
𝜋 𝜋 𝜋
sin ( + 𝜃) = 𝑐𝑜𝑠𝜃 cos ( + 𝜃) = −𝑠𝑖𝑛𝜃 tan ( + 𝜃) = −𝑐𝑜𝑡𝜃
2 2 2
sin(𝜋 − 𝜃) = 𝑠𝑖𝑛𝜃 cos(𝜋 − 𝜃) = −𝑐𝑜𝑠𝜃 tan(𝜋 − 𝜃) = −𝑡𝑎𝑛𝜃
{
sin(𝜋 + 𝜃) = −𝑠𝑖𝑛𝜃 cos(𝜋 + 𝜃) = −𝑐𝑜𝑠𝜃 tan(𝜋 + 𝜃) = 𝑡𝑎𝑛𝜃
3𝜋 3𝜋 3𝜋
sin ( − 𝜃) = −𝑐𝑜𝑠𝜃 cos ( − 𝜃) = −𝑠𝑖𝑛𝜃 tan ( − 𝜃) = 𝑐𝑜𝑡𝜃
{ 2 2 2
3𝜋 3𝜋 3𝜋
sin ( + 𝜃) = −𝑐𝑜𝑠𝜃 cos ( + 𝜃) = 𝑠𝑖𝑛𝜃 tan ( + 𝜃) = −𝑐𝑜𝑡𝜃
2 2 2
sin(2𝜋 − 𝜃) = −𝑠𝑖𝑛𝜃 cos(2𝜋 − 𝜃) = 𝑐𝑜𝑠𝜃 tan(2𝜋 − 𝜃) = −𝑡𝑎𝑛𝜃
{
sin 2(𝜋 + 𝜃) = 𝑠𝑖𝑛𝜃 cos(2𝜋 + 𝜃) = 𝑐𝑜𝑠𝜃 tan(2𝜋 + 𝜃) = 𝑡𝑎𝑛𝜃
For Exe # 10.3 & 10.4

Single Angle Identities: Double Angle Identities:

𝛼 𝛼
• 𝑠𝑖𝑛𝛼 = 2𝑠𝑖𝑛 𝑐𝑜𝑠 • sin 2𝛼 = 2𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼
2 2
𝛼 𝛼 𝑐𝑜𝑠 2 𝛼− 𝑠𝑖𝑛2 𝛼
𝑐𝑜𝑠 2 − 𝑠𝑖𝑛2 2 • cos 2𝛼 = { 2𝑐𝑜𝑠 2 𝛼 − 1
2
𝛼 1 − 2𝑠𝑖𝑛2 𝛼
• cos 𝛼 = 2𝑐𝑜𝑠 2 − 1
2 2𝑡𝑎𝑛 𝛼
𝛼 • tan 2𝛼 =
1− 2𝑠𝑖𝑛2 1−𝑡𝑎𝑛2 𝛼
{ 2
Half Angles Identities

Triple Angle Identities: 𝑺𝒊𝒏

𝜶
= ±√
𝟏 − 𝒄𝒐𝒔 𝜶
𝟐 𝟐
• 𝑠𝑖𝑛3𝛼 = 3𝑠𝑖𝑛𝛼− 4𝑠𝑖𝑛3 𝛼
𝜶 𝟏 + 𝒄𝒐𝒔 𝜶
• 𝑐𝑜𝑠3𝛼 = 4𝑐𝑜𝑠 3 𝛼 − 3𝑐𝑜𝑠𝛼 𝑪𝒐𝒔
𝟐
= ±√
𝟐
3𝑡𝑎𝑛𝛼− 𝑡𝑎𝑛3 𝛼
• tan 3𝛼 = 𝒕𝒂𝒏
𝜶
= ±√
𝟏 − 𝒄𝒐𝒔 𝜶
1−3𝑡𝑎𝑛2𝛼 𝟐 𝟏 + 𝒄𝒐𝒔 𝜶

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 5 of 11
Sum, Difference & Product:
• 2 sin 𝛼 𝑐𝑜𝑠𝛽 = sin(𝛼 + 𝛽 ) + sin (𝛼 − 𝛽) Product
• 2 cos 𝛼 𝑠𝑖𝑛𝛽 = sin(𝛼 + 𝛽 ) − sin (𝛼 − 𝛽) TO
• 2 cos 𝛼 𝑐𝑜𝑠𝛽 = cos(𝛼 + 𝛽 ) + cos (𝛼 − 𝛽) Sum/ Difference
• −2 sin 𝛼 𝑠𝑖𝑛𝛽 = cos(𝛼 + 𝛽 ) − cos (𝛼 − 𝛽)
𝑃+𝑄 𝑃−𝑄
• sin 𝑃 + 𝑠𝑖𝑛𝑄 = 2 sin ( ) 𝑐𝑜𝑠 ( )
2 2
𝑃+𝑄 𝑃−𝑄 Sum/ Difference
• sin 𝑃 − 𝑠𝑖𝑛𝑄 = 2 cos ( ) 𝑠𝑖𝑛 ( )
2 2 TO
𝑃+𝑄 𝑃−𝑄
• cos 𝑃 + 𝑐𝑜𝑠𝑄 = 2 cos ( 2 ) 𝑐𝑜𝑠 ( 2 ) Product
𝑃+𝑄 𝑃−𝑄
• cos 𝑃 − 𝑐𝑜𝑠𝑄 = −2 sin ( ) 𝑠𝑖𝑛 ( )
2 2

CH # 11
Trigonometry Function and their Graph
Period of Trigonometric Function:
“The smallest +ve number which when added to the original circular measure of the
angle gives same value of function is called period.” 𝑖. 𝑒 𝜃 ± 2𝑛𝜋

Period of Trigonometric Function:

• The period of 𝑺𝒊𝒏𝒙 is 𝟐𝝅. • The period of 𝑺𝒆𝒄𝒙 is 𝟐𝝅.
• The period of 𝑪𝒐𝒔𝒆𝒄𝒙 is 𝟐𝝅. • The period of 𝑻𝒂𝒏𝒙 is 𝝅.
• The period of 𝑪𝒐𝒔𝒙 is 𝟐𝝅. • The period of 𝑪𝒐𝒕𝒙 is 𝝅.
CH # 12
Solution of Right Angled Triangle:
In order to solve a right angled triangle, we have to find:
i. The measure of two acute angles. (Less than 𝟗𝟎𝟎 )
ii. The lengths of three sides.

Note:
• Angles are 𝜶, 𝜷, 𝜸.
• Sides are a, b, c.
𝒂 𝒃 𝒂
• 𝒔𝒊𝒏𝜶 = 𝒄 𝒄𝒐𝒔𝜶 = 𝒄 𝒕𝒂𝒏𝜶 =
𝒃

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 6 of 11
Important Formulas of Right Angle Triangle:
i. 𝛼 + 𝛽 + 𝛾 = 180𝑜
ii. 𝛼 + 𝛽 + 90𝑜 = 180𝑜

In (ii) 𝜶 𝑎𝑛𝑑 𝜷 are acute angle (< 90).

Pythagorean / Pythagoras Theorem: (Only Applicable for Right Triangle)

• (Hyp )2 = (𝐵𝑎𝑠𝑒)2 + (𝑃𝑒𝑟𝑝)2

• (H)2 = (𝐵)2 + (𝑃)2
• (c )2 = (𝑎)2 + (𝑏)2

Solution of Oblique Triangle(Which is not right)

Case – I: Case – III:
When one side and two angles are given
• Use Law of Sine When Three side are given
Case – II: • Use Law of Cosine
When Two side and their included angle are given
• First Law of Cosine Then Law of sine.
• First Law of Tangent Then Law of sine.

The Law of Sine:

𝑎 𝑏 𝑐
• = =
𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛽 𝑠𝑖𝑛𝛾
The Law of Cosine:
𝑏 2 +𝑐 2 −𝑎2
• 𝑎 2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 𝑐𝑜𝑠𝛼 • 𝐶𝑜𝑠𝛼 =
2𝑏𝑐
• 𝑏 2 = 𝑐 2 + 𝑎 2 − 2𝑐𝑎 𝑐𝑜𝑠𝛽 𝑐 2 +𝑎2 −𝑏 2
• 𝐶𝑜𝑠𝛽 =
• 𝑐 2 = 𝑎 2 + 𝑏 2 − 2𝑎𝑏 𝑐𝑜𝑠𝛾 2𝑎𝑐
𝑎2 +𝑏 2 −𝑐 2
• 𝐶𝑜𝑠𝛾 = 2𝑎𝑏
The Law of Tangent:
𝛼−𝛽 𝛽−𝛾 𝛾−𝛼
𝑎−𝑏 𝑡𝑎𝑛( 2 ) 𝑏−𝑐 𝑡𝑎𝑛( 2 ) 𝑐−𝑎 𝑡𝑎𝑛( 2 )
• = 𝛼+𝛽 • = 𝛽+𝛾
• = 𝛾+𝛼
𝑎+𝑏 𝑡𝑎𝑛( 2 ) 𝑏+𝑐 𝑡𝑎𝑛( 2 ) 𝑐+𝑎 𝑡𝑎𝑛( 2 )

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 7 of 11
Half Angle Formulas:
𝜶 (𝑠−𝑏)(𝑠−𝑐) 𝜶 (𝑠)(𝑠−𝑎) 𝜶 (𝑠−𝑏)(𝑠−𝑐)
• 𝑆𝑖𝑛 𝟐 = √ • 𝐶𝑜𝑠 𝟐 = √ • 𝑇𝑎𝑛 𝟐 = √
𝑏𝑐 𝑏𝑐 (𝑠)(𝑠−𝑎)

𝜷 (𝑠−𝑐)(𝑠−𝑎) 𝜷 (𝑠)(𝑠−𝑏) 𝜷 (𝑠−𝑐)(𝑠−𝑎)

• 𝑆𝑖𝑛 𝟐 = √ • 𝐶𝑜𝑠 𝟐 = √ • 𝑇𝑎𝑛 𝟐 = √
𝑐𝑎 𝑐𝑎 (𝑠)(𝑠−𝑏)

𝜸 (𝑠−𝑎)(𝑠−𝑏) 𝜸 (𝑠)(𝑠−𝑐) 𝜸 (𝑠−𝑎)(𝑠−𝑏)

• 𝑆𝑖𝑛 𝟐 = √ • 𝐶𝑜𝑠 𝟐 = √ • 𝑇𝑎𝑛 𝟐 = √
𝑎𝑏 𝑎𝑏 (𝑠)(𝑠−𝑐)

Are of Triangle: (= ∆)
Case – I:
When 2 sides and 1 angle ( included )are given
1 1 1
• ∆= 2 𝑏𝑐 𝑠𝑖𝑛𝛼 = 2 𝑐𝑎 𝑠𝑖𝑛𝛽 = 2 𝑎𝑏 𝑠𝑖𝑛𝛾
Case – II:
When 1 side and 2 angles are given
𝑎2 𝑠𝑖𝑛𝛽𝑠𝑖𝑛𝛾 𝑏 2 𝑠𝑖𝑛𝛾𝑠𝑖𝑛𝛼 𝑐 2 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽
• ∆= = =
2𝑠𝑖𝑛𝛼 2𝑠𝑖𝑛𝛽 2𝑠𝑖𝑛𝛾
Case – III:
Hero’s Formula. When 3 sides are given.
• ∆= √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
Circum Radius: (=R) Note:
• 𝑅 = 2𝑠𝑖𝑛𝛼 =
𝑎 𝑏
=
𝑐 • 2s = a + b + c
2𝑠𝑖𝑛𝛽 2𝑠𝑖𝑛𝛾 a+b+c
𝑎𝑏𝑐 • s =
• 𝑅= 2
4∆ Note:
In Radius: (=r) 2∆
∆ • 𝑠𝑖𝑛𝛼 = 𝑏𝑐
• 𝑟=
𝑠 2∆
Note ∆ = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
2 • 𝑠𝑖𝑛𝛽 = 𝑎𝑐
• ∆ = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) 2∆
• 𝑠𝑖𝑛𝛾 =
𝑎𝑏
Escribed Circle: Note:
∆
• 𝑟1 = R Circum Radius
𝑠−𝑎
∆ r In Radius
• 𝑟2 = 𝑟1 , 𝑟2 , 𝑟3 Escribed Radius
𝑠−𝑏
∆
• 𝑟3 = ∆ Area of Triangle
𝑠−𝑐

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 8 of 11
CH # 13
Inverse Trigonometric Function
Inverse Trigonometric Function
Function Domain Range
𝜋 𝜋 𝜋 𝜋
𝑦 = 𝑆𝑖𝑛 −1 𝑥 −1 ≤ 𝑥 ≤ 1 𝑂𝑅 [−1, 1] − ≤𝑦≤ 𝑂𝑅 𝑦 ∈ [− , ]
2 2 2 2
𝑦 = 𝐶𝑜𝑠 −1 𝑥 −1 ≤ 𝑥 ≤ 1 𝑂𝑅 [−1, 1 0 ≤ 𝑦 ≤ 𝜋 𝑂𝑅 𝑦 ∈ [0, 𝜋]
𝜋 𝜋 𝜋 𝜋
𝑦 = 𝑇𝑎𝑛−1 𝑥 R 𝒐𝒓 (−∞ , ∞) −
2
<𝑦<
2
𝑂𝑅 𝑦 ∈ (− , )
2 2
𝜋 𝜋
𝑦 = 𝐶𝑜𝑠𝑒𝑐 −1𝑥 R 𝒐𝒓 (−∞ , ∞) − ≤𝑦≤
2 2
𝑂𝑅 𝑦 ≠ 0

𝑦 = 𝑆𝑒𝑐 −1 𝑥 𝑥 ≥ −1 𝑜𝑟 𝑥 ≤ 1 0 ≤ 𝑦 ≤ 𝜋 𝑂𝑅 𝑦 ∈ [0, 𝜋]

𝑦 = 𝐶𝑜𝑡 −1 𝑥 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1 0 < 𝑦 < 𝜋, 𝑦 ∈ (0, 𝜋)

Note (For MCQ’S):

𝜋 𝜋
i. 𝑆𝑖𝑛−1 𝑥 = 2 −𝐶𝑜𝑠 −1 𝑥 and 𝐶𝑜𝑠 −1 𝑥 = 2 −𝑆𝑖𝑛−1 𝑥
𝜋 𝜋
ii. 𝑇𝑎𝑛−1 𝑥 = 2 −𝐶𝑜𝑡 −1 𝑥 and 𝐶𝑜𝑡 −1 𝑥 = 2 −𝑇𝑎𝑛−1 𝑥
𝜋 𝜋
iii. 𝑆𝑒𝑐 −1 𝑥 = 2 −𝐶𝑜𝑠𝑒𝑐 −1 𝑥 and 𝐶𝑜𝑠𝑒𝑐 −1 𝑥 = 2 −𝑆𝑒𝑐 −1 𝑥

Exe # 13.2
Inverse Trigonometric Formulas:
• 𝑠𝑖𝑛 −1 𝐴 + 𝑠𝑖𝑛−1 𝐵 = 𝑠𝑖𝑛−1 (𝐴√1 − 𝐵2 + 𝐵√1 − 𝐴2 )
• 𝑠𝑖𝑛 −1 𝐴 − 𝑠𝑖𝑛−1 𝐵 = 𝑠𝑖𝑛−1 (𝐴√1 − 𝐵2 − 𝐵√1 − 𝐴2 )
• 𝑐𝑜𝑠 −1 𝐴 + 𝑐𝑜𝑠 −1 𝐵 = 𝑐𝑜𝑠 −1 (𝐴𝐵 − √(1 − 𝐴2 )(1 − 𝐵2 ) )
• 𝑐𝑜𝑠 −1 𝐴 − 𝑐𝑜𝑠 −1 𝐵 = 𝑐𝑜𝑠 −1 (𝐴𝐵 + √(1 − 𝐴2 )(1 − 𝐵2 ) )
𝐴+𝐵
• 𝑡𝑎𝑛−1 𝐴 + 𝑡𝑎𝑛−1 𝐵 = 𝑡𝑎𝑛−1 ( )
1−𝐴𝐵
𝐴−𝐵
• 𝑡𝑎𝑛−1 𝐴 − 𝑡𝑎𝑛−1 𝐵 = 𝑡𝑎𝑛−1 (1+𝐴𝐵)
2𝐴
• 2𝑡𝑎𝑛−1 𝐴 = 𝑡𝑎𝑛−1 (1−𝐴2)
Note:
2𝑡𝑎𝑛𝜃
𝑠𝑖𝑛2𝜃 =
1 + 𝑡𝑎𝑛2 𝜃
Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 9 of 11
CH # 14
Trigonometric Equations:
The equations containing at least one trigonometric function are
called Trigonometric Equation.

Quadrants
I II III IV
0+𝜃 𝜋−𝜃 𝜋+𝜃 2𝜋 − 𝜃
Note:
Also need to remember trigonometric function value table. (From CH#9)

Solution of Trigonometric Equations:

Trigonometric Equations
Equations Solution (𝒙 = )
𝒔𝒊𝒏𝒙 = 𝟎
𝒕𝒂𝒏𝒙 = 𝟎 𝑛𝜋
𝝅
𝒄𝒐𝒔𝒙 = 𝟎 (𝟐𝒏 + 𝟏)
𝟐
𝒔𝒊𝒏𝟐 𝒙 = 𝒔𝒊𝒏𝟐 𝜶
𝒄𝒐𝒔𝟐 𝒙 = 𝒄𝒐𝒔𝟐 𝜶 𝒏𝝅 ± 𝜶
𝒕𝒂𝒏𝟐 𝒙 = 𝒕𝒂𝒏𝟐 𝜶
𝒔𝒊𝒏𝒙 = 𝒔𝒊𝒏𝜶 𝒏𝝅 + (−𝟏)𝒏 𝜶
𝒄𝒐𝒔𝒙 = 𝒄𝒐𝒔𝜶 𝟐𝒏𝝅 ± 𝜶
𝒕𝒂𝒏𝒙 = 𝒕𝒂𝒏𝜶 𝒏𝝅 ± 𝜶
Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 10 of 11

Principal Trigonometric Functions

Functions Domain Range
𝝅 𝝅
𝒚 = 𝒔𝒊𝒏𝒙 − ≤𝒙≤ −𝟏 ≤ 𝒚 ≤ 𝟏
𝟐 𝟐
𝝅 𝝅
𝒚 = 𝒔𝒊𝒏−𝟏 𝒙 −𝟏 ≤ 𝒚 ≤ 𝟏 − ≤𝒙≤
𝟐 𝟐
𝒚 = 𝒄𝒐𝒔𝒙 𝟎≤𝒙≤𝝅 −𝟏 ≤ 𝒚 ≤ 𝟏

𝒚 = 𝒄𝒐𝒔−𝟏 𝒙 −𝟏 ≤ 𝒚 ≤ 𝟏 𝟎≤𝒙≤𝝅
𝝅 𝝅
𝒚 = 𝒕𝒂𝒏𝒙 − <𝒙<
𝟐 𝟐 R
𝝅 𝝅
𝒚 = 𝒕𝒂𝒏−𝟏 𝒙 R − <𝒙<
𝟐 𝟐
𝒚 = 𝒄𝒐𝒕𝒙 𝟎<𝒙<𝝅 R
𝒚 = 𝒄𝒐𝒕−𝟏 𝒙 R 𝟎<𝒙<𝝅
𝝅
𝒚 = 𝒔𝒆𝒄𝒙 [𝟎, 𝝅], 𝒙 ≠ 𝒚 ≤ −𝟏 𝒐𝒓 𝒚 ≥ 𝟏
𝟐
𝝅
𝒚 = 𝒔𝒆𝒄−𝟏 𝒙 𝒙 ≤ −𝟏 𝒐𝒓 𝒙 ≥ 𝟏 [𝟎, 𝝅], 𝒚 ≠
𝟐
𝝅 𝝅
𝒚 = 𝒄𝒐𝒔𝒆𝒄𝒙 [− , ] , 𝒙 ≠ 𝟎 𝒚 ≤ −𝟏 𝒐𝒓 𝒚 ≥ 𝟏
𝟐 𝟐
𝝅 𝝅
𝒚 = 𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙 𝒙 ≤ −𝟏 𝒐𝒓 𝒙 ≥ 𝟏 [− , ] , 𝒚 ≠ 𝟎
𝟐 𝟐
Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144
 Mathematics Notes [Trigonometry] for F.Sc (Pre-Engineering) / ICS – I
Page 11 of 11

Domains, Ranges & Periods of Trigonometric Function

Functions Domains Ranges Periods
𝑦 = 𝑠𝑖𝑛𝑥 −∞ < 𝑥 < +∞ −1 ≤ 𝑦 ≤ +1 2𝜋
𝑦 = 𝑐𝑜𝑠𝑥 −∞ < 𝑥 < +∞ −1 ≤ 𝑦 ≤ +1 2𝜋
𝑦 = 𝑡𝑎𝑛𝑥 𝝅 𝜋
−∞ < 𝑥 < +∞ 𝒊𝒇 𝒙 ≠ (𝟐𝒏 + 𝟏) −∞ ≤ 𝑦 ≤ +∞
𝟐
𝑦 = 𝑐𝑜𝑠𝑒𝑐𝑥 −∞ < 𝑥 < +∞ 𝒊𝒇 𝒙 ≠ 𝒏𝝅 −1 ≥ 𝑦 𝑜𝑟 𝑦 ≥ +1 2𝜋
𝑦 = 𝑠𝑒𝑐𝑥 𝝅 2𝜋
−∞ < 𝑥 < +∞ 𝒊𝒇 𝒙 ≠ (𝟐𝒏 + 𝟏) −1 ≥ 𝑦 𝑜𝑟 𝑦 ≥ +1
𝟐
𝑦 = 𝑐𝑜𝑡𝑥 −∞ < 𝑥 < +∞ 𝒊𝒇 𝒙 ≠ 𝒏𝝅 −∞ ≤ 𝑦 ≤ +∞ 𝜋

Prepared By: Prof. Ali Raza Khan BS (Mathematics) / M.Sc (IT) / B.ed PH # 0321:5359144