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| Trigonometry |
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| With trigonometry we can find the height of a building or the width of a river without actually climbing or crossing. Certain basic definitions are necessary to further develop this subject. The ratios of two sides of a triangle are taken. There are six possible combinations. Each ratio is given a special name. |
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| In this subject we usually use the Greek letters. |
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(i)  (Alpha) (ii) b (Beta) (iii) q (Theta) |
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| (iv) g (Gamma) (v) f (Phi) (vi) l (Lamda) |
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| and so on to indicate the measure of an angle. |
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Let us call  as q. |
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| The side opposite to q is the side BC. |
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| The side adjacent to q is the side AB. |
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| The hypotenuse of the DABC is the side AC. |
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| Now, let us write down the trigonometrical ratios with the help of the above triangles: |
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| (i) Sine q: It is defined as the ratio of the side opposite to q and the hypotenuse. |
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i.e.,  |
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In short we write sin  |
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| (ii) Cosine q: It is defined as the ratio of an adjacent side to q and the hypotenuse, |
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In short,  |
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| (iii) Tangent q : It is defined as the ratio of the side opposite to q and the adjacent side, |
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In short,  |
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Similarly three more ratios can be obtained by taking the reciprocals of  . |
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(iv) Cosecant q is the reciprocal of sin  |
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| It is written as cosec q, |
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  |
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| (v) Secant q is the reciprocal of cos q. |
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| It is written as sec q, |
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  |
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| (vi) Cotangent q is the reciprocal of tan q. |
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| It is written as cot q, |
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| (i) Sin q means a particular ratio. It is not sin multiplied by q. |
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| (ii) For trigonometrical ratios, short form T-ratios will be used. |
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| (iii) In right angled triangles T-ratios are obtained only for acute angles. |
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| (iv) T-ratios depend on only the magnitude of the angles and not on the size of the triangle. |
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| Calculate the T-ratios of angle A and angle C from the given figure. |
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| Find from the given figure, |
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| (i) sin Q |
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| (ii) sin b |
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| (iii) cos R |
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| (iv) cos a |
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In  |
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(i)  |
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(ii)  |
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In  |
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| PR2 = 32 + 62 (by Pythogoras theorem) |
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| PR2 = 9 + 36 |
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| PR2 = 45 |
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(iii)  |
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(iv)  |
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| From the above observation, we find |
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| (i) sin A = cos B where A + B = 90o |
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| \ Sine of an angle = Cosine of its complement |
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| sin A = cos (90 - A)o |
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| sin 40o = cos (90 - 40)o |
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| = cos 50o |
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(ii)  |
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| = tan A |
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| (iii) By Pythagoras theorem |
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| a2 + b2 = c2 |
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| \ sin2A + cos2A = 1 |
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| These three results are called identities. These results hold good for any value of angle A. An identity is true for all values of the unknown. |
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