Page 237 - Class-11-Physics-Part-1_Neat

APPENDICES 211

cos α + cos β

1 1
= 2 cos (a + b ) cos (a b )
2 2
cos α – cos β

1 1
= -2 sin (a + b ) sin ( a b )
Fig. A 5.3 2 2
Mathematical Signs and Symbols
Binomial Theorem
= equals
≅ equals approximately n nx n(n -1)x 2 2
~ is the order of magnitude of (1– x) =1– 1! + 2! +.....(x <1)
≠ is not equal to
≡ is identical to, is defined as 2
> is greater than (>> is much greater than) (1– x) =1m nx + n(n+1)x +.....(x <1)
2
-n
< is less than (<< is much less than) 1! 2!
≥ is greater than or equal to (or, is no less
than) Exponential Expansion
≤ is less than or equal to (or, is no more 2 3
x
than) e =1+ x + x + x +.....
± plus or minus 2! 3!
∝ is proportional to Logarithmic Expansion
∑ the sum of
x or <x > or x the average value of x
av
Trigonometric Identities
0
sin (90 – θ ) = cos θ Trigonometric Expansion
(θθ θθ θ in radians)
0
cos (90 – θ ) = sin θ
sin θ/ cos θ = tan θ

2
2
sin θ + cos θ =1
2
2
sec θ – tan θ = 1
2
2
csc θ – cot θ = 1

sin2 θ = 2 sin θ cos θ
Products of Vectors

2
2
2
cos2 θ = cos θ – sin θ = 2cos θ –1
Let be unit vectors in the x, y and z
2
= 1– 2 sin θ
directions. Then
sin(α ± β ) = sin α cos β ± cos α sin β
cos (α ± β ) = cos α cos β ∓ sin α sin β
tan (α ± β ) = Any vector a with components a , a , and a z
y
x
along the x,y, and z axes can be written,
1 1
sin α ± sin β = 2 sin (a ± ) cosb (a m ) b
2 2

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