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(iv) cosecant A =, (v) secant A =, , (v) cotangent A = ——“8__ = & (v) cotangent C =, , cosecant A = ——, secant A =, , Vk Academy, 10th Trigonometry Notes, , Position of a point P in the Cartesian plane with respect to co-ordinate axes is represented, by the ordered pair (x, y)., , Trigonometry is the science of relationships between the sides and angles of a right-angled, triangle., , Trigonometric Ratios: Ratios of sides of right triangle are called trigonometric ratios., Consider triangle ABC right-angled at B. These ratios are always defined with respect to, acute angle ‘A’ or angle 'C., , If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric, ratios of an angle can be easily determined., , How to identify sides: Identify the angle with respect to which the t-ratios have to be, calculated. Sides are always labelled with respect to the ‘6’ being considered, , Let us look at both cases:, , A, ul %, 3, ‘ £ in, ER) UNS, Os, Cc pt N, Opposite side Adjacent side, Perpendicular Base, Casel: ZA=0 Case Il: ZC =6, , In a right triangle ABC, right-angled at B. Once we have identified the sides, we can define six tRatios with respect to the sides., , casel case Il, ) sineiA a Breenaiowar _. Bo i) sine C= Pemendiculor _. AB, (I) sineA= hypotenuse ~ AC (i) sine C = hypotenuse ~ AC, , _ tase _ AB ) cosine C = vase. — BO, (il) cosine A = hypotenuse ~ AC {il} cosine G = hypotenuse ~~ AC, : _ perpendicular _ BC ‘i = Perpendicular _ AB, (ii) tangent A= Per = {ill tangent C= Se = “ae, hypotenuse _ AC hypotenuse _ AC, , perpendicular — BC (iv) cosecant C =, , perpendicular — AB, hypotenuse _ AG, , hypotenuse __ AC, base ~ AB, , (v) secant C = tase BO, , base AB base. — Be, , perpendicular BC perpendicular ~~ "AB, , Note from above six relationships:, , 1 1, , cosineA', , , , eat, Sind’ cotangent A = Tan!
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However, it is very tedious to write full forms of t-ratios, therefore the abbreviated notations are:, sine Ais sinA, , cosine Ais cos A, , tangent Ais tanA, , cosecant A is cosec A, , secant Ais secA, , cotangent A is cot A, , TRIGONOMETRIC IDENTITIES, , An equation involving trigonometric ratio of angle(s) is called a trigonometric identity, if it is true, for all values of the angles involved. These are:, , _ sind, tan 8 = 039, , _ cos, cot = sind, , e sin? 6 + cos? 8 = 1 => sin? 6 = 1 — cos? 6 > cos? 6 = 1 — sin? 6, , « cosec? 8 — cot? 8 = 1 > cosec? 6 = 1 + cot? 8 > cot” 6 = cosec? 8 - 1, « sec? 6 — tan? 6 = 1 > sec? 8 = 1+ tan? 8 > tan? 6 = sec? 6-1, , « sin@ cosec 8 = 1 => cos @sec8=1=> tan @cotO=1, , ALERT:, A t-ratio only depends upon the angle ‘6’ and stays the same for same angle of different sized, , right triangles., c, , LA, , Value of t-ratios of specified angles:, , ZA a 30° 45° 60° 90°, , ‘ 1 a v3, sinA 0 2 va 2 1, , v3 ms 1, cos A 1 > Va 3 0, tanA 0 4 il v3 not defined, cosec A not defined 2 v2 a 1, secA 1 Za v2 2 not defined, +, , cotA not defined v3 1 Va 0
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The value of sin 6 and cos 6 can never exceed 1 (one) as opposite side is 1. Adjacent side can, never be greater than hypotenuse since hypotenuse is the longest side in a right-angled A., , ‘t-RATIOS' OF COMPLEMENTARY ANGLES, A, , B Cc, , If AABC is a right-angled triangle, right-angled at B, then, ZA+ ZC =90°['. ZA+ ZB+ ZC = 180° angle-sum-property], or ZC = (90° - ZA), , Thus, 2A and ZC are known as complementary angles and are related by the following, relationships:, , sin (90° -A) = cos A; cosec (90° — A) = secA, , cos (90° — A) = sin A; sec (90° — A) = cosec A, , tan (90° — A) = cot A; cot (90° - A) = tanA
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Revision Notes for Ch 8 Introduction to Trigonometry Class, 10th Mathematics, , Trigonometry and Trigonometric Ratios, , — Trigonometry is the branch of mathematics which deals with measurements of angles and, the problems related to relationship between sides and angles., , Trigonometric Ratios:, — It is the ratios of two sides of a right angle triangle., , ~ Suppose we have a right angled triangle ABC., c, , , , , , , , dejnoipuadiag, , , , , , , , ~ Thus the trigonometric Ratios are defined as the largest side of a right angle triangle is known, as hypotenuse., , Also, by Pythagoras theorem, we know that h? =b2+p2,, , Here, 8 is pronounced as theta, ZA=8. We write trigonometric ratios according to angle 8., , , , Trigonometric | Abbreviations | Ratio, , , , , , , , , , , , , , , , , , , , , , , , , , ratios, , : _Hypotenuse, , — rerpendiciar |S] Reciprocal, Secant@ | Sec@ Tispocenase ——y, Cosecant @ Cosec@ arena <
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Trigonometric Ratios for Few Particular Angels, , Trigonometric Ratios of some specific Angles: 0%, 30°, 45%, 60", & 90", , , , , , Note:- Some Angles like cosecé for 6=0° is not defined. We know that cosecé = 1/siné =1/0.1/0 is, not defined hence cosec0° is not defined., , Trigonometric Identities, , ~ Identities are equations which are valid for any conditions. Trigonometric identities are, equations involving trigonometric ratios., , Some Trigonometric Identities are:, , (i) sin?e+cos2e=1, , Sind _, Cos6, , (iv) sec2e-tan2e=1, (v) cosec?e-cot?e=1, , Trigonometric Ratios for Complementary Angles, , (i) sin (90 - 6) = cose, , (ii) cos (90 - @) = sine, , (iii) tan (90 - 8) = cote, (iv) cot (90 - 6) = tane, (v) sec (90 - 8) = cosece, (vi) cosec (90 - 6) = sec
