Reduce the following to the sine or cosine of one angle:
(i) sin145*cos75 - cos145*sin75
(ii) cos35*cos15 - sin35*sin15
Use the formulae:
sin(a+b)= sin(a) cos(b) + cos(a)sin(b)
and
cos(a+b)= cos(a)cos(b) - sin(a)sin)(b)
(1)The quantity = sin(145-75)
= sin 70
= cos 20
note that If a+b=90 degrees then
sin a = cos b
(2) The quantity = cos (35 + 15)
= cos 50
= sin 40
To reduce the given expressions to the sine or cosine of one angle, we can use the trigonometric identities mentioned.
For expression (i), we can rewrite it as:
sin(145)*cos(75) - cos(145)*sin(75)
Using the identity sin(a+b) = sin(a) cos(b) + cos(a) sin(b), we get:
sin(145-75) = sin(70)
This is equal to cos(20) because if a + b = 90 degrees, then sin(a) = cos(b).
Therefore, the reduced form of expression (i) is cos(20).
For expression (ii), we can rewrite it as:
cos(35)*cos(15) - sin(35)*sin(15)
Using the identity cos(a+b) = cos(a) cos(b) - sin(a) sin(b), we get:
cos(35+15) = cos(50)
This is equal to sin(40).
Therefore, the reduced form of expression (ii) is sin(40).
In summary:
(i) sin(145)*cos(75) - cos(145)*sin(75) reduces to cos(20).
(ii) cos(35)*cos(15) - sin(35)*sin(15) reduces to sin(40).