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Introduction To Trigonometry
- 1. Contents 1. Introduction 2.Trigonometric Ratios 3.Trigonometric Ratios Of Some Specific Angles 4.Trigonometric Ratios Of Complementary Angles 5.Trigonometric Identities
- 2. Introduction • In this chapter, we will studysome ratios of thesides of a right triangle with respect to its acuteangles, called “TRIGNOMETRICRATIOSOF THEANGLE”. • We will also define the trigonometric ratios for theangles of measures 0° and 90°. We will calculatetrigonometricratios for some specificangles and establish some identities involving these ratios, called “TRIGNOMETRIC IDENTITIES”.
- 4. Trigonometric Ratios • Let us consider a right triangle, Here angle A is an acute angle. Note the position of the side BC with respect to angle A. we will call the side opposite to angle A. AC is the hypotenuse of the right angled ∆le and the side AB is the part of angle A. so, we will call it as the side adjacent to angle A. • The trigonometric ratios of angle A in right angle ABC are defined as follows: • Sine of angle A= Side opposite to angle A =BC = P Hypotenuse AC H • Cosine of angle A= Side adjacent to angle A= AB = B Hypotenuse AC H • Tangent of angle A= Side opposite to angle A= BC = P Side adjacent to angle A = AB B
- 5. Trigonometric Ratios Cont. • Cosecant of angle A= hypotenuse = AC = H Side opposite to angle A BC P • Secant of angle A = hypotenuse = AC = H Side of adjacent to angle A AB B • Cotangent of angle A = Side adjacent to angle A = AB = B Side opposite to angle A BC P • The ratios defined above are abbreviated as sin A, Cos A, tan A, Cosec A, sec A, and cot A respectively. Note that the ratios Cosec A, Sec A and cot A are respectively, the reciprocals of the ratios Sin A , Cos A, and Tan A. • Also, observe that tan A= BC/AC = Sin A and cot A = Cos A AB/AC Cos A Sin A • So, the trigonometric ratios of an acute angle in a right ∆le express The relationship between the angle and the length of its
- 6. Let Us Try An Example : • QUESTION : given Tan A = 4/3, find the other Trigonometric ratios of the angle A. • SOLUTION : let us first draw a right ∆ ABC now we know that Tan A = BC = 4 = Opp. AB 3 Adj. Therefore, to find hypotenuse we use Pythagoras theorem, AC²= = = = 5 cm Now, we can write all the trigonometric ratios using their definitions. Sin A = BC = 4 AC 5 Cos A = AB = 3 AC 5 Therefore, Cot A = 3 , Cosec A = 5 , and Sec A = 5 4 4 3 ACAB 916 25 ACAB
- 7. Trigonometric Ratios Of Some Specific Angles Trigonometric Ratios Of 45 ° • In ∆le ABC ,right-angled at B, if one angle is 45 °, then other angle is also 45 °. • So , BC = AB • Now , suppose BC = AB = a. • Then , by Pythagoras theorem , AC²= BC² = a²+ a² = 2a² • Using definitions : Sin 45° = BC = a = 1 AC a∫͞₂ ∫͞₂ • Cos 45° = AB = a = 1 AC a∫͞₂ = ∫͞₂ • Tan 45° = BC = a = 1 AB a • Also, cosec 45° = 1 = ∫͞₂, sec 45° = ∫͞₂, cot 45° = 1 = 1 sin 45° tan 45°
- 8. Trigonometric Ratios Of 30° and 60° • Consider an equilateral ∆le ABC. Since each angle is equal to 60°. • So, AB = BC = AC = 60°. • Now, ∆ABC ‗̴ ∆ ACD ( AD is a median and median divides a ∆ into two equal parts). • Therefore, BD = DC and /͟ BAD = /͟ CAD ( CPCT ). • Now, ∆ ABD is a right angled triangle with /͟ BAD =30° and /͟ ABD = 60°. • As we know, for finding friends the trigonometric ratios, we need to know the lengths of the side of the ∆. So, let us suppose that AB = 2a.
- 9. Trigonometric Ratios Of 30°and 60°Cont…. • Then , BD = ½ BC = a And AD²= AB² - BD² = (2a)² - (a)² = 3a², Now we have : Sin 30°= BD = a = , Cos 30°= AD = a∫͞₃ = ∫͞₃ , Tan 30°= BD = a = 1 AB 2a AB 2a 2 AD a∫͞₃ ∫͞₃ Also, Cosec 30° = 1 = 2, sec 30° = 1 = 2 , cot 30° =1 = ∫͞₃ sin 30° Cos 30° ∫͞₃ Similarly, Sin 60°=AD= a∫͞₃ = ∫͞₃, Cos 60°= 1, tan 60°= ∫͞₃ AB 2a 2 2 Cosec 60° = 2 , Sec 60°= 2 and cot 60°= 1 ∫͞₃ ∫͞₃
- 10. Trigonometric ratios of 0°and 90° • Let us see what happens to the trigonometric ratios of angle A, if it is made smaller and smaller in the right triangle ABC, till It becomes zero. As ∠ A gets smaller and smaller, the length of the side BC decreases . The point C gets closer to point B, and finally when ∠ A becomes very close to 0°, AC becomes almost the same as AB.
- 11. Trigonometric Ratios of 0 and 90 Cont. • When ∠ A is very close to 0°, BC gets very close to 0 and so the value of sin A =BC/AC is very close to 0.Also, when ∠ A is very close to 0°, AC is nearly the same as AB and so the value of Cos A =AB/AC is very close to 1. • This helps us to see how we can define the values of Sin A and Cos A when A = 0°. We define : sin 0° = 0 and Cos 0° = 1. • Sin 0° = 0 and Cos 0 ° = 1. • Tan θ ° = Sin θ °= 0, Cot θ °= 1 , which is not defined. Cos θ ° Tan θ °
- 12. Trigonometric Ratios of 0 and 90 Cont. • Sec θ °= 1 = 1 and cosec θ °= 1 , which is again not defined Cos θ ° Sin θ ° • Now, let us see what happens to the trigonometric ratios of ∠ A, when it is made larger and larger in ΔABC till it becomes 90°. As ∠ A gets larger and larger, ∠ C gets smaller and smaller. Therefore, as in the case above, the length of the side AB goes on decreasing. The point A gets closer to point B . Finally when ∠ A is very close to 90°, ∠ C becomes very close to 0° and the side AC almost coincides with side BC.
- 13. Trigonometric Ratios of 0° and 90° Cont….. • When ∠ C is very close to 0°, ∠ A is very close to 90°, side AC is nearly the, • Same as side BC, and so sin A is very close to1. Also when ∠ A is very close to 90°, • ∠ C is very close to 0°, and the side AB is nearly zero, so Cos A is very close to 0. • So, we define : sin 90°= 1 and Cos 90°= 0.
- 14. Trigonometric Ratios Of Some Specific Angles Table…..
- 15. Trigonometric Ratios Of Complementary Angles • As we know that in ∆ABC, right-angled at B, we can see2 complementary angles. • Since /͟ A + /͟ C= 90°, they form such a pair. We have– • Sin A = BC , Cos A = AB ,Tan A = BC , AC AC AB • Cosec A = AC, Sec A = AC, Cot A = AB __ ① BC AB BC
- 16. Trigonometric Ratios Of Complementary Angles Cont. • Now let us write the trigonometric ratios for ∠ C = 90° – ∠ A. For convenience, we shall write 90° – A instead of 90° – ∠ A. What would be the side opposite and the side adjacent to the angle 90° - A ? You will find that AB is the side opposite and BC is the side adjacent to the angle90° – A. Therefore, Sin (90° – A) = AB , cos (90° – A) = BC, tan (90° – A) = AB AC AC BC Cosec (90°-A ) = AC, sec (90°-A ) = AC, cot (90°-A ) = BC __② AB BC AB • Now, compare the ratios in (1) and (2). Observe that : • Sin (90°-A ) = AB = Cos A and Cos (90°-A ) = BC =
- 17. Trigonometric Ratios Of Complementary Angles Cont. • Also , tan (90°-A ) = AB = Cot A , Cot (90°-A) = BC = Tan A. BC AB • Sec (90°-A) = AC = Cosec A, Cosec (90°-A) = AC = Sec A BC AB • So, Sin (90° – A) = Cos A, Cos (90° – A) = Sin A, • Tan (90° – A) = Cot A, Cot (90° – A) = Tan A, • Sec (90° – A) = Cosec A, Cosec (90° – A) = Sec A, • Or all values of angle A lying between 0° and 90°. Check whether this holds for A = 0° or A = 90°. • Note : Tan 0° = 0 = Cot 90°, Sec 0° = 1 = Cosec 90°
- 18. Trigonometric Identities• In Δ ABC, right-angled at B, we have: AB²+BC² = AC² (1) • Dividing each term of (1) by AC² , we get AB² + BC² = AC² AC² AC² AC² I . e., (AB/AC)²+(BC/AC)²=(AC²/AC²) I . e., (Cos A)²+(sin A)²= 1 I . e. , Cos²A+ sin²A= 1 (2) Therefore Cos²A = 1- sin²A And sin²A = 1- Cos²A This is true for all A such that 0° ≤ A ≤ 90°. So, this is a trigonometric identity.
- 19. Cont….. • AB² +BC² = AC² AB² AB² AB² or, (AB/AB)² +(BC/AB)²=(AC/AB)² i.e., 1+tan²A = sec²A (3) Therefore tan²A = 1- sec²A . Is this equation true for A = 0°? Yes, it is. What about A = 90°?Well, tan A and sec A are not defined for A = 90°. So, (3) is true for all A such that 0° ≤ A < 90°.Let us see what we get on dividing (1) by BC². We get AB²+BC²=AC² BC² BC² BC² i.e., (AB²/BC²)+(BC²/BC²)=(AC²/BC²) i.e., cot²A+1 =cosec²A (4) Therefore cot²A = 1- cosec²A.
- 20. How to use Identities• Note that cosec A and cot A are not defined for A = 0°. Therefore(4) is true for all A such that 0° < A ≤ 90°. • 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 us see how we can do this using these identities. Suppose we know that • Tan A= 1, then Cot A=∫͞₃ ∫͞₃ • Since, Sec²A = 1 + Tan²A = 1+1 =4, Sec =2 and Cos A=∫͞₃ 3 3 ∫͞₃ 2 • Again SinA =∫͞͞͞͞͞͞1-͞co͞ s²͞A=∫͞͞͞1-͞3 =1 Therefore, cosec A = 2 ∫ 4 2
- 21. Made By Abhay Vansil Parth Choudhary
- 22. Thank You Hope You Liked It And Also Gained Some Knowledge Out Of It..