Class name Subject Book title Chapter number Content Created at
Class XI Mathematics Mathematics 3 : y ∈ R, y ≤ – 1or y ≥ 1}. The domain of y = tan x is the set {x : x ∈ R and π x ≠ (2n + 1) , n ∈ Z} and range is the set of all real numbers. The ...
Class XI Mathematics Mathematics 3 0 < x < 2 and Reprint 2026-27 54 MATHEMATICS π assumes arbitraily large positive values as x approaches to . Similarly, to say that cosec x decreas...
Class XI Mathematics Mathematics 3 in second quadrant, find the values of other five trigonometricfunctions. 5 12 Solution Since cot x = 12 , we have tan x = 5 2 2 144 169 Now sec x ...
Class XI Mathematics Mathematics 3 connection are called trigonometric identities. We have seen that 1. sin (– x) = – sin x 2. cos (– x) = cos x We shall now prove some more results:...
Class XI Mathematics Mathematics 3 values of x and y in the identities 3, 4, 7 and 8, we get the followingresults: cos ( + x) = – sin x sin ( + x) = cos x 2 2 cos (π– x) = – cos x si...
Class XI Mathematics Mathematics 3 =cot y+cot x Reprint 2026-27 TRIGONOMETRIC FUNCTIONS 61 Since, none of the x, y and (x + y) is multiple of π, we find that sin x sin y and sin (x +...
Class XI Mathematics Mathematics 3 x2 x ≠ nπ + , where n is an integer We have sin (x + y) = sin x cos y + cos x sin y Replacing y by x, we get sin 2x = 2 sin x cos x. 2sinxcosx Agai...
Class XI Mathematics Mathematics 3 an integer We have tan 3x =tan (2x + x) 2tan x 2 + tan x tan 2x + tan x 1– tan x = = 2tan x .tan x 1– tan 2x tan x 1– 2 1– tan x Reprint 2026-27 TR...
Class XI Mathematics Mathematics 3 = θ+φ  and y  θ−φ   2   2  Substituting the values of x and y in (3), (4), (7) and (8), we get  θ+φ  θ−φ  cos θ + cos φ = 2 cos  cos ...
Class XI Mathematics Mathematics 3 π π 5π π L.H.S. = 3sin sec −4sin cot 6 3 6 4 1 π− π  π = 3 × 2 × 2 – 4 sin 6 × 1 = 3 – 4 si6 = 3 – 4 × = 1 = R.H.S. Example 11 Find the value o...
Class XI Mathematics Mathematics 3 + x+ cos − x= 2 cosx  4   4  Solution Using the Identity 20(i), we have Reprint 2026-27 66 MATHEMATICS  π  π  L.H.S. = cos  4 + x+ cos...
Class XI Mathematics Mathematics 3  tan  π + x 2  4  1 tan x  cos (π+ x) cos (− x) 2 7.  π  = 1 tan x  8.  π  = cot x tan  − x   sin (π− x) cos  + x  4   2  3...
Class XI Mathematics Mathematics 3 x 1 6 tan x+ tan x 25. cos 6x = 32 cos x – 48cos x + 18 cos x – 1 Miscellaneous Examples 3 12 Example 18 If sin x = , cos y =− , where x and y both...
Class XI Mathematics Mathematics 3   2 2  = 2 −2sin 2 sin 2             5x 5x = −sin5x sin− 2  = sin5x sin2 = R.H.S. π Example 20 Find the value of tan . So...
Class XI Mathematics Mathematics 3 of the following : 2 2 2 4 1 8. tan x = − , x in quadrant II 9. cos x = − , x in quadrant III 3 3 10. sin = , in quadrant II Summary fi If in a circ...
Class XI Mathematics Mathematics 3 x, y and (± y) is an odd multiple of , then tan (x + y) = tan x + tan y 1− tanxtan y tan x − tan y fi tan (x – y) =1+ tanxtan y fi If none of the ang...
Class XI Mathematics Mathematics 3 Mathematicians, Aryabhatta (476), Brahmagupta (598), Bhaskara I (600) and Bhaskara II (1114) got important results. All this knowledge first went f...
Class XI Mathematics Mathematics 4 76 MATHEMATICS Chapter 4 COMPLEX NUMBERS AND QUADRATIC EQUATIONS v Mathematics is the Queen of Sciences and Arithmetic is the Queen of Mathematics....
Class XI Mathematics Mathematics 6 box can be chosen in 3 different ways. Hence, there are 2×3=6pairsofschoolbagandatiffin box. For each of these pairs a water bottlecanbechosenin2di...
Class XI Mathematics Mathematics 4 have 4x + i (3x – y) = 3 + i (–6) ... (1) Equating the real and the imaginary parts of (1), we get 4x = 3, 3 x – y = – 6, 3 33 which,onsolvingsimul...
Class XI Mathematics Mathematics 4 4.3.2 Difference of two complex numbers Given any two complex numbers z and 1 z2, the difference z1– z 2s defined as follows: z1– z 2 z +1(– z ).2 ...
Class XI Mathematics Mathematics 4 (vi) The distributive law For any three complex numbers z , z , z , 1 2 3 (a) z 1z 2 z )3= z z1+ 2 z 1 3 (b) (z + z ) z = z z + z z 1 2 3 1 3 2 3 4...
Class XI Mathematics Mathematics 4 4.3.6 The square roots of a negative real number Note that i = –1 and ( – i) = i = – 1 Therefore, the square roots of – 1 are i, – i. However, by t...
Class XI Mathematics Mathematics 4 + z ) = (z + z ) (z + z ), 1 2 1 2 1 2 = (z 1 z )2z +1(z + 1 ) z2 2 (Distributivelaw) = z1+ z 2 1 z z1 2z 2 (Distributivelaw) = z + z z + z z + z 2...
Class XI Mathematics Mathematics 4 – 198i. Example 4 Express ( − 3 + −2 2 3)(i )in the form of a + ib Solution We have, (− 3 + −2 )( 2 3 −i ) (= − 3 + 2i )(2 3 −i ) 2 −6+ 2 + 3 1+2 2...