Trigonometry-ratios of angles

1. Sep 12, 2012

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.:(

2. Sep 12, 2012

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.

3. Sep 15, 2012

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.

4. Sep 16, 2012

Celluhh

Oh ok thank you !!

5. Sep 17, 2012

Celluhh

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

6. Sep 17, 2012

dextercioby

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

7. Sep 17, 2012

Studiot

Did you have a problem with my general formulae?

8. Sep 18, 2012

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 !!

9. Sep 18, 2012

Celluhh

Um wait what is C1 ,C2 etc...

10. Sep 18, 2012

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.

11. Sep 18, 2012

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 ?

12. Sep 18, 2012

GingerLee

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

13. Sep 18, 2012

Studiot

These are the binomial coefficients also written

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

They are normally studied before trigonometry.