# Trigonometry-ratios of angles

Celluhh
is there a relation between the numerical answers of cos5A and cosA?

sin4A and sinA?

i want to work backwards, if it is possible. tried deriving a formula by myself, but couldnt.:(

Studiot
$$\begin{array}{l} \sin na = {}^n{C_1}{\cos ^{n - 1}}\sin a - {}^n{C_3}{\cos ^{n - 3}}a{\sin ^3}a + {}^n{C_5}{\cos ^{n - 5}}a{\sin ^5}a...... \\ \cos na = {\cos ^n}a - {}^n{C_2}{\cos ^{n - 2}}a{\sin ^2}a + {}^n{C_4}{\cos ^{n - 4}}a{\sin ^4}a............ \\ \end{array}$$

Where a is the angle and n an integer.

GingerLee
I think, you can use the sum of two angles approach

Sin4A = 2 Sin2A Cos2A
= 2 (2 SinA CosA) Cos2A
= 4 SinA CosA (Cos²A - Sin²A)
= 4 SinA CosA (1 - 2Sin²A)
= 4 CosA (SinA - 2 Sin³A)
= 4 √(1 - Sin²A)(SinA - 2 Sin³A)

Similar approach can be taken for other one.

Celluhh
Oh ok thank you !!

Celluhh
What about for fractions ? For example sin1/3 x?

Homework Helper
For fractions it's essentially not doable, except for n=2,3,4, because of the algebra involved.

Studiot
Did you have a problem with my general formulae?

Celluhh
@studiot, no that's not it but it's hard to memorise it and it's not one of the formulas learnt in school for
Now , so I can't exactly use it in my exam ! Thanks a lot though !!

Celluhh
Um wait what is C1 ,C2 etc...

GingerLee
They are symbols for combination. Also written as C(n,1).
If you have not studied permutations, combinations, factorial yet, then you wont understand them.

Celluhh
Oh I see yep I'm only at the double angle formulae level ... And having problems with expressing cos4a or others in the form of simple trigo ratio eg. Cosa. Does anyone have any online website to recommend that solves this kind of problems ?

GingerLee
Have you ever heard of wolframalpha? I am not sure if I should post links in this forum, but you can google it.

Studiot
These are the binomial coefficients also written

$$\left( {\begin{array}{*{20}{c}} n \\ r \\ \end{array}} \right)$$

They are normally studied before trigonometry.