Plane trigonometry, by S.L. Loney.

ADDITION AND SUBTRACTION FORMULE.

91.

93. Ex.. Find the values of sin 75~ and cos1150.
sin 75~ sin (45~ + 30~) =sin 45~ cos 30~ + cos 45~ sin 30~
1 7/3 1 1 _/3 +1
->/'2 ~ +'2- 2 2/2'
and cos 750 = cos (450 + 30~) = cos 45~ cos 30~ - sin 45~ sin 30~
- 1 V3 1 1 V/3-1,/2 2 J2 2- 2J2
Ex. 2. Assuming the formulae for sin ( + y) and cos (x +y), deduce
the formul foor sin (x - y) and cos (x - y).
We have
sin x sin {(x - y)+ y}=sin(x - y) cos y + os (x - ) sin y......(1),
and cosx=cos {(x - y) + y}=cos ( - y) cos y - sin (x -y) siny...... (2).
Multiplying (1) by cos y and (2) by sin y and subtracting, we have
sin x cos y - cos x sin y = sin (x - y) {cos2 y + sin2 y } = sin (x -- y).
Multiplying (1) by sin y and (2) by cos y and adding, we have
sin x sin y + co cos y = os (x - y) { os2 y +sin2 y} = cos (x - y).
Hence the two formulae required are proved.
These two formulae are true for all values of the angles since the
formulae from which they are derived are true for all values.
EXAMPLES. XIII.
3 9
1. If sin a = 5 and cos I = 41 find the value of sin (a - 3) andcos (a + ).
45 33
2. If sina= 5 and sin=-6, find the values of sin (a - ) and
sin (a + ).
15 12
3. If sin a = 1 and cos P = 13, find the values of sin (a + /), cos (a - 3)
and tan (a + f).
Prove that
4. sin (A + B) sin (A - B) =sin2 A - sin2 B.
5. cos (A+B) cos(A - B)=os A - sin2 B.
6. cos (45~ - ) cs (45~ -B)- sin (45~ -A) sin (45~ -B)= sin (A +B).