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Introduction to, Trigonometry, , Quick Revision, Trigonometric Ratios (iv) cosecant A or cosec A, The ratios of the sides of a right angled triangle = Hypotenuse [i« 4) i AC, with respect to its acute angles, are called Side opposite to 2A P}) BC, , trigonometric ratios. (v) secant A or sec A, , Trigonometric ratios are also called T-ratio. Hypotenuse _ HY) AC, Trigonometric ratios of ZA in right angled AABC ~ Side adjacent to ZA ( 2 } “AB, are defined below. J, , B, , (vi) cotangent A or cot A, , , , , , , , , , Ae _ Side adjacent to 2A (i 3)- AB, a 5 ~ Side opposite to ZA\ P ~ BC, 2 8 Similarly trigonometric ratios of ZC are, Be AB BC, Ae asin = b)eos C=, ae, ,_ AB , AC, Aside adjacent? (c) tan C= BC (d) cosecC = AB, to ZA [ie. Base (B)] a Pe, A ‘ (e) sec C = —— (f) cot C = =~, (i) sine A or sin 4 = SSS pppoe eA Li =) BC AB, Hypotenuse A c, _ BC J &, AC & {|g, (ii) cosine A or cos A & go, _ Side adjacent to 2A (ic 3) _ AB & / aa, Hypotenuse H AC y gs, (iii) tangent A or tan A A B, 8, , Side opposite to ZC, , _ Side opposite to ZA [ie ‘) BC [ie Perpendicular (P)], , © Side adjacent to “A

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116, , CBSE New Pattern ~ Mathematics X (Term-l), , , , A Popular Technique to Remember, , , , . .:. PBP, T-ratios i.e., HHB, Pandit (P) Badari(B) Prasad (P ), Har (H) Har (H ) Bholay (B), x, H Pp, z 5 Y, Then, an@=—. coo - 4, nin, H H B, = cosec@= A sec@ = A cot 8 = 3, , where, P is perpendicular, B is base and H is, hypotenuse., , Important Points, , (i) In an isosceles right AABC, right angled, at B, the trigonometric ratios obtained by, taking either A or £C, both give the, same value., , (i) The value of each of the trigonometric, ratios of an angle does not depend on, the size of the triangle. It only depends, on the angle., , (iii) It is clear that the values of the, trigonometric ratios of an angle do not, vary with the lengths of the sides of the, triangle, if the angle remains the same., , , , If one of the trigonometric ratios of an, , acute angle is known, then remaining, , trigonometric ratios of that angle can be, , determined easily., , (v) Each trigonometric ratio is a real number, and has no unit., , (vi) As, the hypotenuse is the longest side in, , a right angled triangle, the value of, , sin A or cos Ais always less than 1, , (or in particular equal to 1) whereas the, , value of sec 4 or cosec A is always, , greater than or equal to 1., , Relation Between Trigonometric Ratios, , , , , , , , , , (i? sin A= ,cosec A= a, cosec A sin A, (id cos A= L , secA= i, sec A cos A, (ii?) tan A= L scot A= i, cot A tan A, (iv) tan A= ain, cos A, (reo A= S84, sin A, , Values of Trigonometric Ratios for, Some Specific Angles, , , , , , , , , , Angles 0° 30° 45° 60° 90°, sin 0 0 1 1 WB 1, 2 v2 2, cos 1 v3 1 & 0, 2 v2 a, tan® 0 1 1 V3 i, V3, cosec® © 2 va 1, __ ih, secO 1 2 V2 2 oo, v3, cot 9 ce 13 1 ae 0, V3, , Here, c¢ = undefined, , Important Points, , (i) The value of sin 6 increase from 0 to 1 andcos®, decrease from 1 to 0, where 0< 0< 90°., , (i? In the case of tan @, the values increase from 0 to ©,, where 0<0<90°., , (iii) In the case of cot 0, the values decrease from ~ to 0,, where 0 <6 < 90°., , {iv} In the case of cosec @, the values decrease from - to, 1, where 0 < 8< 90°., , (v) In the case of sec, the values increase from 1 to ~,, where 0<@<90°., , (vi! Division by 0 is not allowed, since 1/0 is, indeterminate (not defined).

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CBSE New Pattern ~ Mathematics X (Term-I) 117, , , , Trigonometric Identity (ii) sec? @ — tan? @ =1 or 1 + tan? = sec? 0, , An equation is called an identity when it is true for or sec’ @—1=tan2@, , all values of the variables involved. Similarly, an . z 5 ss, , equation involving trigonometric ratios of an angle (iii) 1+ cot" @ = cosec”@ orcosec” 8 — cot” O=1, , is called a trigonometric identity, if it is true for all or cot? =cosec26 -1, , values of the angle involved. For any acute angle 6, » > 2 9, , we have Note sin“@ = (sin @)° but sin 9° # (sin6}”. The same, (i) sin? 8 + cos? =1 or sin?@=1-cos?@ is true for all other trigonometric ratios., , or cos’ =1— sin?@, , Representation of a Trigonometric Ratio in Terms of Any Other Trigonometric Ratio, , , , , , , , , , , , , , , , , , , , , , sin 8 cosO tan 6 cot 6 sec 0 cosec@, =a tan@ I 2 1, ‘ sin 8 1-cos?8) j———— - —————__ (sec - 1), sind Yarian®®) Ytrcot?6) cece coseed, [a=sin?0) cos0 1 cotd 1 Kcosec 6-1), y(l-sin” 6) TT ——sS x, cosé yil+tan?0) — (1+cot? 6) wee cosec @, 8 a = 1, tang ee tan 8 (sec? 6-1), a i Bonner ake, y(l—sin*) cos cot ylcosec” 8 -1), 8 (1-sin” 6) a 1 cot 1. scosec” 0 -1, eo sin® y(1—cos? 8) tan® (sec? @~1), te 1 y+ tan?0) (0 +c0t? 0) secO __cosec, sec (1 sin26) cos er y(cosec? 6 — 1), 9 2s 1 fittan?9) (a+cor29) = _see__ cosec 8, SORSe) sin 0 yli-cos’0) ~ ane ylsec? 6-1), Objective Questions, Multiple Choice Questions (a)3/5 (b) 3/4, 5, 1. Trigonometry is branch of Mathematics {c]4/s (ars, in which we deal with the relationship 3. If sind = z then cos® is equal to, between angle and sides of a triangle. INCERT Exemplar], (a) True (b) False ei 5 (b)°, (c) Cannot say (d) Partially True/False yb? a? a, , 4, 2. Ifcos A = Fe then the value of tan A is, . INCERT Exemplar] 5

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118, , CBSE New Pattern ~ Mathematics X (Term-l), , , , 10., , . Ifcos® = * then the value of, , 2sec7@ + 2tan? @ —9 is, (a)2 (b}-2, (c)1 (d)3, , . If xcos A =8 and L5cosec A =8 sec A,, , then the value of x is, (a)20, (c)17, , (b) 16, (d) 13, , . Ifasin A=5 and 7cosecA =6 sec A,, , then the value of o is, , 46 46, a, b, ( iS (b) 7, 49, , (c) —, , (d) None of these, 46, , . If is an acute angle and 4 sin@ = 3,, , then the value of 4 sin?@ - 3cos?6 + is, , 45 36, (aay (eae, 32 47, (oo hee, 4 1-sinA, . If sin A =—, then the value of ————, 5 1+cosA, is, 1 1, zh b)—, (a) ( IS, 3 Ss, (c)2 (de, , . Ina APOR, 2Q =90°. If PQ =10 em, , and PR=15cm. Then, the value of, tan? P + sec*P +1 is, , 5 3, , b, M2 ( le, 4, ee dj—, (ele ( Ie, , In AABG, if AB = BC, ZB = 90°, then, the value sin A is, , jul,, ali, , 1, 2)0 d=, (c) (a), , 11., , 12., , 13., , 14., , 15., , 16., , 17., , 18., , In AABG, if AB = 2BC, ZB = 90°, then, the value of sec A is, , A, BL190° c, 1 v3, laj—— (b)—, v2 2, i, ‘5 1, (co) (d), 2 2, The value of cos6 increase as 6 increase., (a) True (b) False, (c) Cannot say (d) Partially True/False, , ‘ 10, Is tan? 30° + cot? 30° = 3, , (b) False, (d) Partially True/False, , (a) True, (c) Cannot say, , sin 2A = 2sin Ais true when Ais equal to, , , , (a)0° (b) 30°, (c)45° (d) 60°, cot? 30° + cosec 30° + 3 tan? 30° is, equal to, {a)3 (b)1, {c)O0 (d)6, The value fae will be, 1—tan* 30°, {a)cos60° (b)sin60°, (c) tan60° (d)cot 60°, If tan 8 + 1 = 2 then the value of, tan®, cosec® is, 1, 1 b)—, (a) ( 5, v3, 2 d)—, (c)v2 (d) a, , Tf x tan 45° cos 60° = sin 60° cot 60°, then, the value of x will be, , (a) |, , (bo), , 3B, , (c} 2, , (d), , —

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CBSE New Pattern ~ Mathematics X (Term-I) 119, , , , , , , , , , , , , , 19. 7cos 30°+5 tan 30°+6 cot 60° is 99, Thewdive of 1-tan? 45°., . ie value OF, 18, ae ws 1+ tan? 45°, la : (a)tango° (b)1, . iz t, (Qe ay ees (c)sinab® (a)0, aye tan 30°, 20. If tand ane 4 8 = 3, then sin? 0 is 28. The value of cot 60° =, , 3 fell (b)=1, ay Why (oa (a), 2, io ce, 17 7 29. If A, B, Care the angles of a AABC,, 1 1 y, 21. If sina = 5 and cosB = > then the value then the valueor tan (° a: “jis, 2, of (a +B) is, (a) 60° (b) 90° [a}eote (brand, 30° d) 45°, si I " (choot (decor, 22. If tan A= Te and tan B = V3, then 2 2, v3 30. Ifcos A +.cos? A =], then the value of, tan (A +B) is > é, 1 sin* A+sin* Ais, (a)O (b), v3 (a0 (b)1, (c)1 (d} 6 (ce) (d)2, 23. If V3 tan@ = 2sin8, then the value 31. Is sin(A + B) = sin A + sin B?, sin? 6 — cos? 6 is (a) True, 1 i (b) False, alo tel-5 (c) Cannot say, (3 (dy (qd) Partially True/False, 2 2 32. | 1-sinO 19 —tan™'9?, 24. If sin@ — cos =0, then the value of Ho l+sin0 Seer an aE, sin‘ @ +cos* @ will be | [NCERT Exemplar] ay THe, , (, fl (jt ae, (, , 4 2 c) Cannot say, ia (dl d) Partially True/False ,, 4 l+tan° A., , 33. The value of, , 25. If m tan 30° cot60°= sin 45° cos 45°, then tae A 8, the value of m will be fi, 1 3 {alsec*A, tale OS (b)-1, 1 3 (c)cot? A, {ele (iy (d)tar? 4, 26. The value of tan = is 34. The value of (1 + tan 8 + sec0), 1+ tan* 30° (1+ cot —cosec8) is, (a)sin60° (b}cos 60° (a) (b)1, , (c) tan60° (d)sin30° (c)2 (d)-1