Question

Find sin 2a and cot 2a:

Answer 1

`sin(2\alpha )=2((5)/(12))((12)/(13))=(10)/(13)\\cot(2\alpha ) = (16511)/(18720)`

Explanation:

`\text{if } cos(\alpha)=(12)/(13)\\\text{That must mean we have a triangle with base 12, and hypotenuse 13.}\\\text{Using Pythagoras, we can determine the base of the triangle must be 5.}\\a^2+b^2=c^2 \text{, where c is the hypotenuse and a, b are the two other sides.}\\c^2-b^2=a^2\\√(c^2-b^2)=a\\√(13^2-12^2)=√(169-144)=√(25)=5\\\text{Therefore, }sin(\alpha) = (5)/(12)\\sin(2\alpha)=2sin(\alpha )cos(\alpha)\\\text{(From double angle formulae identities)}\\`

`sin(2\alpha )=2((5)/(12))((12)/(13))=(10)/(13)\\cos(2\alpha )=cos^2(\alpha)-sin^2(\alpha)\\cos(2\alpha )=((12)/(13))^2-((5)/(12))^2=(16511)/(24336)\\cot(2\alpha)=(cos(2\alpha))/(sin(2\alpha))=((16511)/(24336))/((10)/(13))=(16511)/(18720)`