__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 sin^{2}a
and once in terms of cos2a**.**

**Solution:** In formula
(2) above that is cos(2a) = cos** ^{2}**a - sin

replacing cos** ^{2}**a by
1-sin

cos2a = 1 - 2sin |

Now, if sin** ^{2}**a is
replaced by
1-cos

cos2a = 2cos |

**Example 2:**
Write once sin** ^{2}**a and once
cos

**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 - 2sin** ^{2}**a)

**
=** sina
- 2sin^{3}a
**+** 2sina
cos** ^{2}**a

**=**
sina - 2sin^{3}a
**+** 2sina
(1 - sin** ^{2}**a)
,or

sin3a
= 3sina
- 4sin^{3}a
(13) |

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

cos3a
= 4cos^{3}a
- 3cosa
(14) Verify. |

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

__D)
cot3a :__ The derivation s left for students

__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**:**

**Exercises:**