Page 26 - DMTH201_Basic Mathematics-1

Unit 2: Trigonometric Functions-II

or sin(A + B) = sin A cos B + cos A sin B .....(II) Notes
For any two numbers A and B
sin(A B) = sin A cos B cos A sin B
Proof: Replacing B by B in (2), we have

sin(A + ( B)) = sin A cos( B) + cos A sin( B)
or sin(A B) = sin A cos B cos A sin B

Example: Find the value of each of the following:
(i) sin5 /12 (ii) cos /12 (iii) cos7 /12

Solution
(a) (i) sin 5 /12 = sin ( /4 + /6) = sin /4. cos /6 + cos /4. sin /6
= 1/√2. 3/2 + 1/ 2.1/2
sin 5 /12 = 3 + 1/ 2. 1/2 = 3 + 1/2 2
(ii) cos /12 = cos( /4 – /6)

= cos /4 . cos /6+ sin /4+ sin /6
= 1/ 2. 3/2+ 1/ 2.1/2 = 3 + 1/2 2
cos /12 = √3 + 1/2/ 2
Observe that sin 5 /12 = cos /12

(iii) cos 7 /12 = cos ( /3 + /4)
= cos /3 . cos /4 sin /3 . sin /4
= 1/2. 1/ 2 √3/2. 1/ 2 = 1 √3 /2 2
cos 7 /12 = 1 3/2 2

2.1.2 Transformation of Products into Sums and Inverse

Transformation of Products into Sums or Differences

We know that
sin(A + B) = sin A cos B + cos A sin B
sin(A B) = sin A cos B cos A sin B

cos(A + B) = cos A cos B sin A sin B
cos(A B) = cos A cos B + sin A sin B
By adding and subtracting the first two formulae, we get respectively

2sin A cos B = sin(A + B) + sin(A B) …(1)

and 2cos A sin B = sin(A + B) sin(A B) …(2)

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