Upcoming SlideShare
×

# นำเสนอตรีโกณมิติจริง

2,702 views

Published on

2 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total views
2,702
On SlideShare
0
From Embeds
0
Number of Embeds
51
Actions
Shares
0
80
0
Likes
2
Embeds 0
No embeds

No notes for slide

### นำเสนอตรีโกณมิติจริง

1. 1. By Nittaya Noinan Kanchanapisekwittayalai phetchabun school Grade 10 Trigonometry
2. 2. <ul><li>Trigonometry is derived from Greek words trigonon (three angles) and metron ( measure). </li></ul><ul><li>Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees </li></ul><ul><li>Triangles on a sphere are also studied, in spherical trigonometry. </li></ul><ul><li>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. </li></ul>Trigonometry
3. 3. History <ul><li>The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago. </li></ul><ul><li>Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books </li></ul><ul><li>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. </li></ul><ul><li>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). </li></ul><ul><li>Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al-Kashi made significant contributions in this field(trigonometry). </li></ul>
4. 4. Right Triangle <ul><li>A triangle in which one angle is equal to 90  is called right triangle. </li></ul><ul><li>The side opposite to the right angle is known as hypotenuse. </li></ul><ul><li>AB is the hypotenuse </li></ul><ul><li>The other two sides are known as legs. </li></ul><ul><li> AC and BC are the legs </li></ul>Trigonometry deals with Right Triangles
5. 5. Pythagoras Theorem <ul><ul><ul><li>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 . </li></ul></ul></ul><ul><ul><ul><li>In the figure </li></ul></ul></ul><ul><ul><ul><ul><li>AB 2 = BC 2 + AC 2 </li></ul></ul></ul></ul>
6. 6. Trigonometry ( 三角幾何 ) means “Triangle” and “Measurement” Introduction Trigonometric Ratios In F.2 we concentrated on right angle triangles .
7. 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. 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. 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. 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. 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. 12. Adjacent , Opposite Side and Hypotenuse of a Right Angle Triangle .
13. 13. Adjacent side Opposite side hypotenuse 
14. 14. hypotenuse Adjacent side Opposite side 
15. 15. Three Types Trigonometric Ratios <ul><li>There are 3 kinds of trigonometric ratios we will learn. </li></ul><ul><ul><li>sine ratio </li></ul></ul><ul><ul><li>cosine ratio </li></ul></ul><ul><ul><li>tangent ratio </li></ul></ul>
16. 16. Sine Ratios <ul><li>Definition of Sine Ratio. </li></ul><ul><li>Application of Sine Ratio. </li></ul>
17. 17. <ul><li>Definition of Sine Ratio . </li></ul>1 If the hypotenuse equals to 1 Sin  =  Opposite sides
18. 18. <ul><li>Definition of Sine Ratio . </li></ul>For any right-angled triangle Sin  =  Opposite side hypotenuses
19. 19. Exercise 1 In the figure, find sin  Sin  = Opposite Side hypotenuses = 4 7  = 34.85  (corr to 2 d.p.)  4 7
20. 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. 21. Cosine Ratios <ul><li>Definition of Cosine. </li></ul><ul><li>Relation of Cosine to the sides of right angle triangle. </li></ul>
22. 22. <ul><li>Definition of Cosine Ratio . </li></ul>1 If the hypotenuse equals to 1 Cos  =  Adjacent Side
23. 23. <ul><li>Definition of Cosine Ratio . </li></ul>For any right-angled triangle Cos  =  hypotenuses Adjacent Side
24. 24. Exercise 3  3 8 In the figure, find cos  cos  = adjacent Side hypotenuses = 3 8  = 67.98  (corr to 2 d.p.)
25. 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. 26. Tangent Ratios <ul><li>Definition of Tangent. </li></ul><ul><li>Relation of Tangent to the sides of right angle triangle. </li></ul>
27. 27. <ul><li>Definition of Tangent Ratio. </li></ul>For any right-angled triangle tan  =  Adjacent Side Opposite Side
28. 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. 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. 30. Conclusion Make Sure that the triangle is right-angled
31. 31. Trigonometric ratios <ul><li>Sine(sin) opposite side/hypotenuse </li></ul><ul><li>Cosine(cos) adjacent side/hypotenuse </li></ul><ul><li>Tangent(tan) opposite side/adjacent side </li></ul><ul><li>Cosecant(cosec) hypotenuse/opposite side </li></ul><ul><li>Secant(sec) hypotenuse/adjacent side </li></ul><ul><li>Cotangent(cot) adjacent side/opposite side </li></ul>
32. 32. Values of trigonometric function of Angle A <ul><li>sin  = a/c </li></ul><ul><li>cos  = b/c </li></ul><ul><li>tan  = a/b </li></ul><ul><li>cosec  = c/a </li></ul><ul><li>sec  = c/b </li></ul><ul><li>cot  = b/a </li></ul>
33. 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. 34. Calculator <ul><li>This Calculates the values of trigonometric functions of different angles. </li></ul><ul><li>First Enter whether you want to enter the angle in radians or in degrees. Radian gives a bit more accurate value than Degree. </li></ul><ul><li>Then Enter the required trigonometric function in the format given below: </li></ul><ul><li>Enter 1 for sin. </li></ul><ul><li>Enter 2 for cosine. </li></ul><ul><li>Enter 3 for tangent. </li></ul><ul><li>Enter 4 for cosecant. </li></ul><ul><li>Enter 5 for secant. </li></ul><ul><li>Enter 6 for cotangent. </li></ul><ul><li>Then enter the magnitude of angle. </li></ul>
35. 35. Trigonometric identities <ul><li>sin 2 A + cos 2 A = 1 </li></ul><ul><li>1 + tan 2 A = sec 2 A </li></ul><ul><li>1 + cot 2 A = cosec 2 A </li></ul><ul><li>sin(A+B) = sinAcosB + cosAsin B </li></ul><ul><li>cos(A+B) = cosAcosB – sinAsinB </li></ul><ul><li>tan(A+B) = (tanA+tanB)/(1 – tanAtan B) </li></ul><ul><li>sin(A-B) = sinAcosB – cosAsinB </li></ul><ul><li>cos(A-B)=cosAcosB+sinAsinB </li></ul><ul><li>tan(A-B)=(tanA-tanB)(1+tanAtanB) </li></ul><ul><li>sin2A =2sinAcosA </li></ul><ul><li>cos2A=cos 2 A - sin 2 A </li></ul><ul><li>tan2A=2tanA/(1-tan 2 A) </li></ul><ul><li>sin(A/2) = ±  {(1-cosA)/2} </li></ul><ul><li>Cos(A/2)= ±  {(1+cosA)/2} </li></ul><ul><li>Tan(A/2)= ±  {(1-cosA)/(1+cosA)} </li></ul>
36. 36. Relation between different Trigonometric Identities <ul><li>Sine </li></ul><ul><li>Cosine </li></ul><ul><li>Tangent </li></ul><ul><li>Cosecant </li></ul><ul><li>Secant </li></ul><ul><li>Cotangent </li></ul>
37. 37. Angles of Elevation and Depression <ul><li>Line of sight: The line from our eyes to the object, we are viewing. </li></ul><ul><li>Angle of Elevation:The angle through which our eyes move upwards to see an object above us. </li></ul><ul><li>Angle of depression:The angle through which our eyes move downwards to see an object below us. </li></ul>