| Class XI |
Mathematics |
Mathematics |
3 |
Chapter 3 TRIGONOMETRICFUNCTIONS v A mathematician knows how to solve a problem, he can not solve it. – MILNE v 3.1 Introduction The word ‘trigonom... |
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| Class XI |
Mathematics |
Mathematics |
3 |
of measurement of an angle which are most commonly used, viz. degree measure and radian measure. 1 th 3.2.1 Degree measure Ifarotationfromtheinitia... |
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| Class XI |
Mathematics |
Mathematics |
3 |
as initial side of an angle. Thenthelength ofanarcofthecirclewillgivetheradian 1 measure of the angle which the arc will subtend at the centre of t... |
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| Class XI |
Mathematics |
Mathematics |
3 |
π Radian measure = 180 × Degree measure Degree measure = π ×Radian measure Example 1 Convert 40° 20′ into radian measure. Solution We know that 180... |
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| Class XI |
Mathematics |
Mathematics |
3 |
radian π 22π and θ 2 110° = ×110 = radian 180 36 Let l be the length of each of the arc. Then l = r θ = r θ , which gives 1 1 2 2 13π 22π 1 22 × r1... |
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| Class XI |
Mathematics |
Mathematics |
3 |
2 2 OM + MP = OP or a + b = 1 Thus, for every point on the unit circle, we have a + b = 1 or cos x + sin x = 1 Since one complete revolution subten... |
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| Class XI |
Mathematics |
Mathematics |
3 |
n is any integer. sinx 1 π sec x = cosx , x ≠ (2n + 12 , where n is any integer. sinx π tan x = , x ≠ (2n +12 , where n is any integer. cosx cosx c... |
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| Class XI |
Mathematics |
Mathematics |
3 |
xb are both positive, in the π second quadrant ( < x <π) a is negative and b is positive, in the third quadrant 3π 3π (π < x < ) a and b are both n... |
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| 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 ... |
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| Class XI |
Mathematics |
Mathematics |
3 |
x increases for 0 < x < 2 and Reprint 2026-27 54 MATHEMATICS π assumes arbitraily large positive values as x approaches to . Similarly, to say that... |
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| Class XI |
Mathematics |
Mathematics |
3 |
, x lies 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 N... |
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| Class XI |
Mathematics |
Mathematics |
3 |
By taking suitable 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 ... |
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| Class XI |
Mathematics |
Mathematics |
3 |
x + y) =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 s... |
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| Class XI |
Mathematics |
Mathematics |
3 |
= 1+ tan 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. 2sinx... |
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| Class XI |
Mathematics |
Mathematics |
3 |
where n is 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... |
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| Class XI |
Mathematics |
Mathematics |
3 |
φ. Therefore x = 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 ... |
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| Class XI |
Mathematics |
Mathematics |
3 |
Solution We have 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... |
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| Class XI |
Mathematics |
Mathematics |
3 |
π π cos + 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 ... |
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| Class XI |
Mathematics |
Mathematics |
3 |
4 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 ... |
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| Class XI |
Mathematics |
Mathematics |
3 |
8sin x cos 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... |
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| Class XI |
Mathematics |
Mathematics |
3 |
2 2 2 2 = 2 −2sin 2 sin 2 5x 5x = −sin5x sin− 2 = sin5x sin2 = R.H.S. π Example 20 Find the value of ta... |
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| Class XI |
Mathematics |
Mathematics |
3 |
of the angles 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 n... |
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| Class XI |
Mathematics |
Mathematics |
3 |
The ancient Indian Mathematicians, Aryabhatta (476), Brahmagupta (598), Bhaskara I (600) and Bhaskara II (1114) got important results. All this kno... |
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| 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.... |
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| 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... |
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