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Trigonometry

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Introduction To Trigonometry
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Trigonometry

This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors

This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors

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Trigonometry

1. 1. INTRODUCTION TO TRIGONOMETRY
2. 2. There is perhaps nothing which so occupies the middle position of mathematics as trigonometry. - J.F. Herbart
3. 3. INDEX • Introduction • Trigonometric ratios • Trigonometric ratios of some specific angles • Trigonometric ratios of complementary angles • Trigonometric identities
4. 4. INTRODUCTION • We will get to know about some ratios of the sides of a right triangle with respect to its acute angles called TRIGNOMETRIC RATIOS OF ANGLES. • We will also define the trigonometric ratios for the angles of measures 0° and 90°. We will calculate trigonometric ratios for some specific angles and establish some identities involving these ratios, called TRIGNOMETRIC IDENTITIES.
5. 5. What is TRIGONOMETRY?? • TRIGONOMETRY is that branch of mathematics which deals with the measurement of the sides and the angles of a triangle. • The word TRIGONOMETRY is derived from Greek words: ‘tri’(meaning three), ‘gon’(meaning sides) and ‘metry’(meaning measure).
6. 6. TRIGONOMETRIC RATIOS • Let us consider a right triangle. • Here angle A is an acute angle. The position of the side BC with respect to angle A is the side opposite to angle A. AC is the hypotenuse of the right angled triangle and the side AB is the part of angle A. So, we call it as the side adjacent to angle A. • The trigonometric ratios of angle A in right angle ABC are defined as follows :
7. 7. • Sin A= side opposite to angle A = BC hypotenuse AC • Cosine A= side adjacent to angle A = AB hypotenuse AC • Tangent A= Side opposite to angle A = BC Side adjacent to angle A AB • Cosecant A= 1 = AC Sine A BC • Secant A= 1 = AC Cosine A AB • Cotangent A= 1 = AB Tangent A BC • The ratios are abbreviated as sin A, cos A, tan A, cosec A, sec A and cot A respectively. • The ratios cosec A, sec A and cot A are the reciprocals of the ratios sin A, cos A, and tan A respectively. • Also, tan A= sin A and cot A= cos A cos A sin A • So, the Trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.
8. 8. Let us take an example.. • If sin A= 3 , find the other trigonometric ratios 5 of the angle A. Solution:- Let a constant term be ‘a’ sin A= perpendicular = BC = 3a hypotenuse AC 5a By Pythagoras Theorem, h²= p² + b² (5a) ²= (3a) ² + b² 25a²= 9a² + b² b²= 25a² – 9a² b²=16a² B= 4a
9. 9. Therefore, cos A= AB = 4a = 4 AC 5a 5 Tan A= BC = 3a = 3 AB 4a 4 Cosec A= AC = 5a = 5 BC 3a 3 Sec A= AC = 5a = 5 AB 4a 4 Cot A= AB = 4a = 4 BC 3a 3
10. 10. TRIGONOMETRIC RATIOS OF SOME SPECIFIC ANGLES
11. 11. TRIGONOMETRIC RATIOS OF COMPLEMENTARY ANGLES • As we know that, ABC is a right Angled triangle, right angled at B, We know, Sin A= BC Cos A= AB Tan A= BC AC AC AB Cosec A= AC Sec A= AC Cot A= AB ______❶ BC AB BC • There is one pair of complementary angle in the above triangle. angleA +angleC = 90 Therefore, C= 90- A AB is the opposite side and BC is the adjacent side to the angle 90- A. So, Sin (90-A)= BC Cos (90-A)= AB Tan (90-A)= BC AC AC AB
12. 12. Cosec (90-A) = AC Sec (90-A) = AC Cot (90-A) =AB __❷ BC AB BC • When we compare the ratios in ❶ and ❷, we get: Sin (90-A)= Cos A Cos (90-A)= Sin A Tan (90-A)= Cot A Cosec (90-A) = Sec A Sec (90-A) = Sec A Cot (90-A) = Tan A
13. 13. TRIGONOMETRIC IDENTITIES • In triangle ABC, right angled at B: AC²=AB²+BC²(By Pythagoras Theorem) Dividing each term by AC², we get AC² = AB² + BC² AC² AC² AC² 1= (Cos A) ²+(Sin A)² [ Cos A= AB and Sin A= BC ] 1=Cos²A + Sin²A ______❶ AC AC This is first trigonometric identity.
14. 14. Again dividing AC²=AB²+BC² BY AB², we get, AC²=AB²+BC² AB² AB² AB² (SecA) ²= 1+(TanA)² Sec²A= 1+ Tan²A_______❷ This is second trigonometric identity. Now, dividing AC²=AB²+BC² BY BC², WE GET, AC²=AB²+BC² BC² BC² BC² (CosecA) ²= (TanA)²+1 Cosec²A= Tan²A+1_______❸ This is third trigonometric identity.
15. 15. • Using these identities, we can express each trigonometric ratio in terms of other trigonometric ratios, i.e. if any one of the ratios is known, we can also determine the values of other trigonometric ratios. • Let’s see how we can do this using these identities. We know that, Tan A=1 and Cot A=∫3 ∫3 And by 2nd identity, we know Sec²A=1+ Tan²A =1+(1/∫3)²= 1+1/3=4/3 Sec A=∫4/3= 2/∫3 Cos A= ∫3/2 Sin²A=1- Cos²A=1-(∫3/2)²= 1-3/4=1/4 Sin A=∫1/4=1/2 Cosec A=2