Chapter 11

The Trig Ratios of Angles 2a and 3a

It is easy to find the trig ratios for double-angle (2a) and triple-angle (3a).  For double angle, any (a+b) can be replaced by (a+a) or 2a as follows:

Example 1:  Write cos2a once in terms of sin2a and once in terms of cos2a.

Solution:  In formula (2) above that is cos(2a) = cos2a - sin2a,   

replacing cos2a by 1-sin2a, we get:          cos2a = 1 - sin2a - sin2a, or

cos2a = 1 - 2sin2a         (5)

Now, if  sin2a is replaced by 1-cos2a, we get:

cos2a = 2cos2a - 1        (6)

Example 2:  Write once sin2a and once cos2a in terms of cos2a.

Example 3: 

The Trig Ratios of 3a

A) sin3a:

sin3a may be written as sin(a + 2a) that is equal to

sin3a = sina cos2a   +   cosa sin2a =

        = sina (1 - 2sin2a) + cosa (2sina cosa)

        = sina - 2sin3a + 2sina cos2a

      = sina - 2sin3a + 2sina (1 - sin2a)    ,or

sin3a = 3sina - 4sin3a     (13)

B) sin3a: In a similar way, it can be shown that

cos3a = 4cos3a - 3cosa    (14) Verify.

C) tan3a: The method is the same as worked out below:

D) cot3a: The derivation s left for students. Show that the result is:

Two Other Important Formulas:

The calculation of [sina] and [cosa] in terms of [tan(a/2)]

The same process may be repeated for cosa as follows: