Question

Question #52e7f

Answer 1

1. See explanation

Explanation:

Using the appropriate `color(blue)"Addition Formulae"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(cos(A±B)=cosAcosB∓sinAsinB)color(white)(a/a)|)))`

hence cos(x-y)=cosxcosy+sinxsiny

We are given sinx and cosy but require to find cosx and siny to evaluate the right side of the expansion.

Using the following sketches (not to scale)

`sinx=3/5 "allows opposite and hypotenuse to be completed"`

Use `color(blue)"Pythagoras' theorem"` to find adjacent = 4 or recognise the well known '3,4,5' triangle.

`sinx=3/5rArrcosx=4/5`
`"------------------------------------------------------"`

Similarly `cosy=5/13` allows adjacent and hypotenuse to be completed. Using Pythagoras or recognising '5,12,13' triangle to obtain opposite = 12.

`cosy=5/13rArrsiny=12/13`
`"-----------------------------------------------------------"`

We now have all the required ratios to complete the expansion.

`cos(x-y)=cosxcosy+sinxsiny`

`=(4/5xx5/13)+(3/5xx12/13)`

`=20/65+36/65=56/65" thus shown"`