Sums and differences of angles
1. sin Ha + bL = sinHaL cosHbL + cosHaL sinHbL
2. sinHa - bL = sinHaL cosHbL - cosHaL sinHbL
3. cos Ha + bL = cosHaL cosHbL - sinHaL sinHbL
4. cos Ha - bL = cosHaL cosHbL + sinHaL sinHbL
5. tan Ha + bL =
6. tan Ha - bL =
tanHaL + tanHbL
1 - tanHaL tanHbL
tanHaL - tanHbL
1 + tanHaL tanHbL
Example 1
Expand and simplify cosH2 a - bL - cosH2 a + bL.
Using formula H4L, cos H2 a - bL = cos H2 aL cos HbL + sin H2 aL sin HbL
Using formula H3L, cos H2 a + bL = cos H2 aL cos HbL - sin H2 aL sin HbL
cos H2 a - bL - cos H2 a + bL =
cos H2 aL cos HbL + sin H2 aL sin HbL - cos H2 aL cos HbL + sin H2 aL sin HbL = 2 sinH2 aL sinHbL
\ cos H2 a - bL - cos H2 a + bL = 2 sin H2 aL sin HbL
Example 2
Without using a calculator, find the exact value of cos105ë .
Hint: Write105ë as a sum of two angles with known ratios.
cos 105ë = cosH60ë + 45ë L =
cosH60ë L cosH45ë L - sinH60ë L sinH45ë L =
\ cos 105ë =
2 4
6
1
1
´
2
3
-
2
1
´
2
1-
3
=
2
1-
3
=
2
2
2
´
2
2
2 =
2
4
6