Trigonometry deals with relationships between sides and angles of triangles, especially right triangles. It has been used for thousands of years in fields like astronomy, navigation, architecture, engineering, and more modern fields like digital imaging and computer graphics. Trigonometric functions define ratios between sides of a right triangle and are used to solve for unknown sides and angles. Common applications include calculating distances, heights, satellite positioning, and modeling waves and vibrations.

1. By Nittaya Noinan Kanchanapisekwittayalai phetchabun school Grade 10 Trigonometry

2. Trigonometry is derived from Greek words trigonon (three angles) and metron ( measure). Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees Triangles on a sphere are also studied, in spherical trigonometry. Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, on the trigonometric functions, and with calculations based on these functions. Trigonometry

3. History The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago. Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly compute the sine of π/4 (45 ° ) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle). Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al-Kashi made significant contributions in this field(trigonometry).

4. Right Triangle A triangle in which one angle is equal to 90  is called right triangle. The side opposite to the right angle is known as hypotenuse. AB is the hypotenuse The other two sides are known as legs. AC and BC are the legs Trigonometry deals with Right Triangles

5. Pythagoras Theorem In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs . In the figure AB 2 = BC 2 + AC 2

6. Trigonometry ( 三角幾何 ) means “Triangle” and “Measurement” Introduction Trigonometric Ratios In F.2 we concentrated on right angle triangles .

7. Unit Circle A Unit Circle Is a Circle With Radius Equals to 1 Unit.(We Always Choose Origin As Its centre) 1 units x Y

8. Trigonometry Trigonometry is the branch of mathematics that looks at the relationship between the length of the sides and the angles in a right angled triangle. It helps us calculate the length of unknown sides, without drawing the triangle, and it also helps us calculate the size of an unknown angle without having to measure it. Let us look at a right angle triangle.

9. Trigonometry O F C Hypotenuse x 0 Opposite Adjacent O F C Hypotenuse y 0 Opposite Adjacent The ‘Hypotenuse’ is always opposite the right angle The ‘Opposite’ is always opposite the angle under investigation. The ‘Adjacent’ is always alongside the angle under investigation.

10. Trigonometry 6 3 O F C Let us look at the ratio or fraction of the opposite over the adjacent. Let us now split this triangle up. 27 0

11. Trigonometry 4 2 2 3 O E F B C Let us look at the ratio or fraction of the opposite over the adjacent Let us split it again. 27 0

12. Adjacent , Opposite Side and Hypotenuse of a Right Angle Triangle .

13. Adjacent side Opposite side hypotenuse 

14. hypotenuse Adjacent side Opposite side 

15. Three Types Trigonometric Ratios There are 3 kinds of trigonometric ratios we will learn. sine ratio cosine ratio tangent ratio

16. Sine Ratios Definition of Sine Ratio. Application of Sine Ratio.

17. Definition of Sine Ratio . 1 If the hypotenuse equals to 1 Sin  =  Opposite sides

18. Definition of Sine Ratio . For any right-angled triangle Sin  =  Opposite side hypotenuses

19. Exercise 1 In the figure, find sin  Sin  = Opposite Side hypotenuses = 4 7  = 34.85  (corr to 2 d.p.)  4 7

20. Exercise 2 11 In the figure, find y Sin35  = Opposite Side hypotenuses y 11 y = 6.31 (corr to 2.d.p.) 35 ° y Sin35  = y = 11 sin35 

21. Cosine Ratios Definition of Cosine. Relation of Cosine to the sides of right angle triangle.

22. Definition of Cosine Ratio . 1 If the hypotenuse equals to 1 Cos  =  Adjacent Side

23. Definition of Cosine Ratio . For any right-angled triangle Cos  =  hypotenuses Adjacent Side

24. Exercise 3  3 8 In the figure, find cos  cos  = adjacent Side hypotenuses = 3 8  = 67.98  (corr to 2 d.p.)

25. Exercise 4 6 In the figure, find x Cos 42  = Adjacent Side hypotenuses 6 x x = 8.07 (corr to 2.d.p.) 42 ° x Cos 42  = x = 6 Cos 42 

26. Tangent Ratios Definition of Tangent. Relation of Tangent to the sides of right angle triangle.

27. Definition of Tangent Ratio. For any right-angled triangle tan  =  Adjacent Side Opposite Side

28. Exercise 5  3 5 In the figure, find tan  tan  = adjacent Side Opposite side = 3 5  = 78.69  (corr to 2 d.p.)

29. Exercise 6 z 5 In the figure, find z tan 22  = adjacent Side Opposite side 5 z z = 12.38 (corr to 2 d.p.) 22  tan 22  = 5 tan 22  z =

30. Conclusion Make Sure that the triangle is right-angled

31. Trigonometric ratios Sine(sin) opposite side/hypotenuse Cosine(cos) adjacent side/hypotenuse Tangent(tan) opposite side/adjacent side Cosecant(cosec) hypotenuse/opposite side Secant(sec) hypotenuse/adjacent side Cotangent(cot) adjacent side/opposite side

32. Values of trigonometric function of Angle A sin  = a/c cos  = b/c tan  = a/b cosec  = c/a sec  = c/b cot  = b/a

33. Values of Trigonometric function 0 30 45 60 90 Sine 0 0.5 1/  2  3/2 1 Cosine 1  3/2 1/  2 0.5 0 Tangent 0 1/  3 1  3 Not defined Cosecant Not defined 2  2 2/  3 1 Secant 1 2/  3  2 2 Not defined Cotangent Not defined  3 1 1/  3 0

34. Calculator This Calculates the values of trigonometric functions of different angles. First Enter whether you want to enter the angle in radians or in degrees. Radian gives a bit more accurate value than Degree. Then Enter the required trigonometric function in the format given below: Enter 1 for sin. Enter 2 for cosine. Enter 3 for tangent. Enter 4 for cosecant. Enter 5 for secant. Enter 6 for cotangent. Then enter the magnitude of angle.

35. Trigonometric identities sin 2 A + cos 2 A = 1 1 + tan 2 A = sec 2 A 1 + cot 2 A = cosec 2 A sin(A+B) = sinAcosB + cosAsin B cos(A+B) = cosAcosB – sinAsinB tan(A+B) = (tanA+tanB)/(1 – tanAtan B) sin(A-B) = sinAcosB – cosAsinB cos(A-B)=cosAcosB+sinAsinB tan(A-B)=(tanA-tanB)(1+tanAtanB) sin2A =2sinAcosA cos2A=cos 2 A - sin 2 A tan2A=2tanA/(1-tan 2 A) sin(A/2) = ±  {(1-cosA)/2} Cos(A/2)= ±  {(1+cosA)/2} Tan(A/2)= ±  {(1-cosA)/(1+cosA)}

36. Relation between different Trigonometric Identities Sine Cosine Tangent Cosecant Secant Cotangent

37. Angles of Elevation and Depression Line of sight: The line from our eyes to the object, we are viewing. Angle of Elevation:The angle through which our eyes move upwards to see an object above us. Angle of depression:The angle through which our eyes move downwards to see an object below us.

38. Problem solved using trigonometric ratios CLICK HERE!

39. Applications of Trigonometry This field of mathematics can be applied in astronomy,navigation, music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

40. Derivations Most Derivations heavily rely on Trigonometry. Click the hyperlinks to view the derivation A few such derivations are given below:- Parallelogram law of addition of vectors . Centripetal Acceleration . Lens Formula Variation of Acceleration due to gravity due to rotation of earth . Finding angle between resultant and the vector.

41. Applications of Trigonometry in Astronomy Since ancient times trigonometry was used in astronomy. The technique of triangulation is used to measure the distance to nearby stars. In 240 B.C., a mathematician named Eratosthenes discovered the radius of the Earth using trigonometry and geometry. In 2001, a group of European astronomers did an experiment that started in 1997 about the distance of Venus from the Sun. Venus was about 105,000,000 kilometers away from the Sun .

42. Application of Trigonometry in Architecture Many modern buildings have beautifully curved surfaces. Making these curves out of steel, stone, concrete or glass is extremely difficult, if not impossible. One way around to address this problem is to piece the surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create what looks like a curved surface. The more regular these shapes, the easier the building process. Regular flat shapes like squares, pentagons and hexagons, can be made out of triangles, and so trigonometry plays an important role in architecture.

43. Waves The graphs of the functions sin(x) and cos(x) look like waves. Sound travels in waves, although these are not necessarily as regular as those of the sine and cosine functions. However, a few hundred years ago, mathematicians realized that any wave at all is made up of sine and cosine waves. This fact lies at the heart of computer music. Since a computer cannot listen to music as we do, the only way to get music into a computer is to represent it mathematically by its constituent sound waves. This is why sound engineers, those who research and develop the newest advances in computer music technology, and sometimes even composers have to understand the basic laws of trigonometry. Waves move across the oceans, earthquakes produce shock waves and light can be thought of as traveling in waves. This is why trigonometry is also used in oceanography, seismology, optics and many other fields like meteorology and the physical sciences.

44. Digital Imaging In theory, the computer needs an infinite amount of information to do this: it needs to know the precise location and colour of each of the infinitely many points on the image to be produced. In practice, this is of course impossible, a computer can only store a finite amount of information. To make the image as detailed and accurate as possible, computer graphic designers resort to a technique called triangulation . As in the architecture example given, they approximate the image by a large number of triangles, so the computer only needs to store a finite amount of data. The edges of these triangles form what looks like a wire frame of the object in the image. Using this wire frame, it is also possible to make the object move realistically. Digital imaging is also used extensively in medicine, for example in CAT and MRI scans. Again, triangulation is used to build accurate images from a finite amount of information. It is also used to build "maps" of things like tumors, which help decide how x -rays should be fired at it in order to destroy it.

45. Conclusion Trigonometry is a branch of Mathematics with several important and useful applications. Hence it attracts more and more research with several theories published year after year Thank You

46. END