Question

How do you use the angle sum identity to find the exact value of cos 105?

Answer 1

Please see the explanation below

Explanation:

Apply

`cos(a+b)=cosacosb-sinasinb`

Therefore,

`cos105=cos(60+45)`

`=cos60cos45-sin60sin45`

`=(1/2)xx(sqrt2/2)-(sqrt3/2)xx(sqrt2/2)`

`=1/4(sqrt2-sqrt6)`

`=0.2588`

Answer 2

`(sqrt2-sqrt6)/4`

Explanation:

`cos(105)`

= `cos(60+45)`

Recall: `cos(a+b)=cosacosb-sinasinb)`
= `cos60cos45-sin60sin45`

= `1/2times1/sqrt2-sqrt3/2times1/sqrt2`

= `1/(2sqrt2)-sqrt3/(2sqrt2)`

common denominator so you can turn the two separate fractions into one fraction
= `(1-sqrt3)/(2sqrt2)`

rationalise the denominator
= `(1-sqrt3)/(2sqrt2)timessqrt2/sqrt2`

simplify
= `(sqrt2-sqrt6)/4`

Answer 3

`1/4(sqrt2-sqrt6)`

Explanation:

`"using the "color(blue)"trigonometric identity"`

`•color(white)(x)cos(x+y)=cosxcosy-sinxsiny`

`cos105=cos(60+45)`

`=cos60cos45-sin60sin45`

`=(1/2xxsqrt2/2)-(sqrt3/2xxsqrt2/2)`

`=sqrt2/4-sqrt6/4=1/4(sqrt2-sqrt6)`