130. to prove Given sin (A + B) = sin A cos B + cos A sin B.........(1), sin (A – B) = sin A cos B – cos A sin B ....(2). Then we have sin D = sin (D – B) cos B + cos (D – B) sin B. But we have shown in the last article that from (1) it follows that and ..cos (D-B) = cos D cos B+ sin D sin B. Substituting for cos (D– B) in above, sin D = sin (D – B) cos B + cos D cos B sin B + sin D sin2 B, .. sin D (1 − sin2 B) = sin (D − B) cos B + cos D cos B sin B, .. sin D cos2 B - cos D cos B sin B = sin (D − B) cos B, Writing A for D we have (2). 131. Given to prove and and sin (A + B) = sin A cos B + cos A sin B..... ...(1), ...(2), (AB) ......(5), (AB) ......(6). (1) + (2) gives sin (A + B) + sin (A – B) = 2 sin A cos B, Put A+ BS and A- B=T, .. (adding) 24 = S+T, (subtracting) 2B = S − T, Writing A for S and B for T we have (5) and (6). · cos A = 2 sin § (A + B) sin 1⁄2 (A – B) .........................(8). 6 132. The signs in the formulæ (1)—(8) may be connected with the fact that the sine increases but the cosine decreases as the angle increases. and 133. In sin (A + B) = sin A cos B + cos A sin B cos (A + B) = cos A cos B - sin A sin B 134. To express the sum or difference of two sines or cosines as a product. This is done in the general case in Arts. 124 and 125. But the following method is recommended as an exercise in the four important fundamental formulæ of Arts. 120, 121, 122, 123. Express the two given angles as the sum and difference, respectively, of two new angles; either by inspection or by solving an equation. Then apply the formulæ of Arts. 120-123. Thus Example 1. sin 7A+ sin 34=sin (5A+2A)+sin (5A − 2A) = (sin 5A cos 24 +cos 5A sin 2A) + (sin 5A cos 2A - cos 5A sin 2A) =2 sin 54 cos 24. Example 2. cos 5A cos 114= cos (84 -34) - cos (84 +34) = (cos 8A cos 34 +sin 84 sin 3A) – (cos 8A cos 3A – sin 84 sin 3A) =2 sin 84 sin 3A. Example 3. sin A-sin B: assuming A to be the greater. Let x+y=A and x-y=B. sin(x+y) -- sin (x − y) = 2 cos x sin y. But solving for x and y, 2x=A+B and 2y=A- B, But solving for x and y, 2x=B+A and 2y=B-A. ...cos A-cos B=2 sin † (B+ A) sin § (B – A). 135. To express the product of sines or cosines as a sum or difference. This is the converse problem to that of the last article. The method is simply to introduce a product containing the ratios complementary to those in the given product. Thus = (sin 44 cos 34 + cos 4A sin 3A) – (sin 44 cos 34 - cos 4A sin 3A) =(cos 7A cos 24+ sin 74 sin 24) + (cos 74 cos 2A - sin 74 sin 24) =cos 5d+cos 94. EXAMPLES V. 1. Show that cos (A – B) + sin (A + B) = (cos A + sin A) (cos B + sin B) cos (AB) sin (A + B) = (cos A – sin A) (cos B - sin B) cos (A+B) sin (AB) = (cos A-sin A) (cos B + sin B). 5. 1cot A tan B If sin A = cos B = 3, find sin (A + B) and cos (A + B). 6. If sin A = 45 and sin B=33, find sin (A – B) and cos (A – B). 7. 8. 9. 53 Find tan (4 + 45°) and cot (45° – A). Show that sin A + cos A = √2. sin (4 + 45°) = √2 cos (45° – A). Show that cos A - sin A = √2 . sin (45° – A) = √√2 cos (45° + A). 10. Find sin 30° from the identity cos 30° = sin 60°. Prove the following identities: 11. (sin A + cos 4)2 = 1 + sin 24. Put the following expressions (17-28) in the form of twice |