Trigonometry functions, Study notes of Mathematics

This document contains the basic trigonometric functions.

Typology: Study notes

2025/2026

Available from 03/16/2026

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Trigonometry
The word trigonometry is derived from the Greek word โ€œtrigonโ€ and โ€œmetronโ€ and it means
measuring the sides of a triangle. Trigonometry was used to solve geometric problems involving
triangles. In the present, it is used in many fields such as,
a) Seismology
b) Designing electric circuits
c) Describing the state of an atom
d) Predicting the heights of tides in the ocean
e) Analyzing a musical tone
The list goes onโ€ฆ
1. Angle
Angle can be measured in clockwise or anticlockwise direction. When measured in clockwise
direction, the angle is negative and angle is positive when measured in anticlockwise direction as
shown in the figure. The ray OA is the initial side and OB is the terminal side in positive angle.
2. Degree and Radian
One full revolution of initial side is 360ยฐ. The angle one degree (1ยฐ) is when the rotation form
initial side to terminal side is (1360
โ„ )๐‘กโ„Ž of one revolution. A degree is divided into 60 minutes
and 1 minute is divided into 60 seconds.
1ยฐ = 60' and 1' = 60"
The relationship between degree and radian is as following:
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Trigonometry

The word trigonometry is derived from the Greek word โ€œtrigonโ€ and โ€œmetronโ€ and it means

measuring the sides of a triangle. Trigonometry was used to solve geometric problems involving

triangles. In the present, it is used in many fields such as,

a) Seismology

b) Designing electric circuits

c) Describing the state of an atom

d) Predicting the heights of tides in the ocean

e) Analyzing a musical tone

The list goes onโ€ฆ

1. Angle

Angle can be measured in clockwise or anticlockwise direction. When measured in clockwise

direction, the angle is negative and angle is positive when measured in anticlockwise direction as

shown in the figure. The ray OA is the initial side and OB is the terminal side in positive angle.

2. Degree and Radian

One full revolution of initial side is 360ยฐ. The angle one degree (1ยฐ) is when the rotation form

initial side to terminal side is (

โ„ )๐‘กโ„Ž of one revolution. A degree is divided into 60 minutes

and 1 minute is divided into 60 seconds.

1ยฐ = 60' and 1' = 60"

The relationship between degree and radian is as following:

3. Trigonometric functions

sin ๐‘ฅ = 0 , ๐‘ฅ = ๐‘›๐œ‹, ๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ ๐‘› ๐‘–๐‘  ๐‘Ž๐‘›๐‘ฆ ๐‘–๐‘›๐‘ก๐‘’๐‘”๐‘’๐‘Ÿ

cos ๐‘ฅ = 0 , ๐‘ฅ =

sin ๐‘ฅ

sec ๐‘ฅ =

cos ๐‘ฅ

tan ๐‘ฅ =

sin ๐‘ฅ

cos ๐‘ฅ

cot ๐‘ฅ =

cos ๐‘ฅ

sin ๐‘ฅ

0ยฐ ฯ€/6 ฯ€/4 ฯ€/3 ฯ€/2 ฯ€ 3ฯ€/2 2ฯ€

sin x 0

cos x 1

tan x 0

Not

defined

Not

defined

4. Sign of trigonometric functions

cos

= cos ๐‘ฅ

sin

= โˆ’ sin ๐‘ฅ

  • cot(๐‘ฅ โˆ’ ๐‘ฆ) =

cot ๐‘ฅ cot ๐‘ฆ+ 1

cot ๐‘ฆ+cot ๐‘ฅ

  • cos 2 ๐‘ฅ = ๐‘๐‘œ๐‘ 

2

2

2

2

1 โˆ’๐‘ก๐‘Ž๐‘›

2

๐‘ฅ

1 +๐‘ก๐‘Ž๐‘›

2

๐‘ฅ

  • sin 2 ๐‘ฅ = 2 sin ๐‘ฅ cos ๐‘ฅ =

2 tan ๐‘ฅ

1 +๐‘ก๐‘Ž๐‘›

2

๐‘ฅ

  • tan 2 ๐‘ฅ =

2 tan ๐‘ฅ

1 โˆ’๐‘ก๐‘Ž๐‘›

2

๐‘ฅ

  • sin 3 ๐‘ฅ = 3 sin ๐‘ฅ โˆ’ 4 ๐‘ ๐‘–๐‘›

3

  • cos 3 ๐‘ฅ = 4 ๐‘๐‘œ๐‘ 

3

๐‘ฅ โˆ’ 3 cos ๐‘ฅ

  • tan 3 ๐‘ฅ =

3 tan ๐‘ฅโˆ’๐‘ก๐‘Ž๐‘›

3

๐‘ฅ

1 โˆ’ 3 ๐‘ก๐‘Ž๐‘›

2

๐‘ฅ

  • cos ๐‘ฅ + cos ๐‘ฆ = 2 ๐‘๐‘œ๐‘  (

๐‘ฅ+๐‘ฆ

2

๐‘ฅโˆ’๐‘ฆ

2

  • cos ๐‘ฅ โˆ’ cos ๐‘ฆ = โˆ’ 2 ๐‘ ๐‘–๐‘› (

๐‘ฅ+๐‘ฆ

2

๐‘ฅโˆ’๐‘ฆ

2

  • sin ๐‘ฅ + sin๐‘ฆ = 2 ๐‘ ๐‘–๐‘› (

๐‘ฅ+๐‘ฆ

2

๐‘ฅโˆ’๐‘ฆ

2

  • sin ๐‘ฅ โˆ’ sin๐‘ฆ = 2 ๐‘๐‘œ๐‘  (

๐‘ฅ+๐‘ฆ

2

๐‘ฅโˆ’๐‘ฆ

2