นำเสนอตรีโกณมิติจริง

By Nittaya Noinan Kanchanapisekwittayalai phetchabun school Grade 10 Trigonometry

[object Object],[object Object],[object Object],[object Object],Trigonometry

History ,[object Object],[object Object],[object Object],[object Object],[object Object]

Right Triangle ,[object Object],[object Object],[object Object],[object Object],[object Object],Trigonometry deals with Right Triangles

Pythagoras Theorem ,[object Object],[object Object],[object Object]

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

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

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.

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.

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

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

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

Adjacent side Opposite side hypotenuse 

hypotenuse Adjacent side Opposite side 

Three Types Trigonometric Ratios ,[object Object],[object Object],[object Object],[object Object]

Sine Ratios ,[object Object],[object Object]

[object Object],1 If the hypotenuse equals to 1 Sin  =  Opposite sides

[object Object],For any right-angled triangle Sin  =  Opposite side hypotenuses

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

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 

Cosine Ratios ,[object Object],[object Object]

[object Object],1 If the hypotenuse equals to 1 Cos  =  Adjacent Side

[object Object],For any right-angled triangle Cos  =  hypotenuses Adjacent Side

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

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 

Tangent Ratios ,[object Object],[object Object]

[object Object],For any right-angled triangle tan  =  Adjacent Side Opposite Side

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

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 =

Conclusion Make Sure that the triangle is right-angled

Trigonometric ratios ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Values of trigonometric function of Angle A ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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

Calculator ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Trigonometric identities ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Relation between different Trigonometric Identities ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Angles of Elevation and Depression ,[object Object],[object Object],[object Object]

Problem solved using trigonometric ratios CLICK HERE!

Applications of Trigonometry ,[object Object]

Derivations ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Applications of Trigonometry in Astronomy ,[object Object],[object Object],[object Object],[object Object]

Application of Trigonometry in Architecture ,[object Object],[object Object],[object Object],[object Object],[object Object]

Waves ,[object Object],[object Object],[object Object],[object Object],[object Object]

Digital Imaging ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Conclusion ,[object Object],Thank You

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