Question

Without using a calculator evaluate `sin45^o/(cos30^o +sin60^o`?

Answer 1

`(sqrt(6))/6`

Explanation:

Soh Cah Toa

Soh `->" sin = " ("opposite")/("hypotenuse")`

Cah `->"cos = "("adjacent")/("hypotenuses")`

Toa `-> "tan = "("opposite")/("adjacent")`

Tony B

Given: `(sin(45^o))/(cos(30^o)+sin(60^o))`

Note that `sin(45^o)=1/sqrt(2)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider the `cos(30^o)+sin(60^o)`

Looking at the diagram we have

`cos(30^o)+sin(60^o)=(sqrt(3 ))/2+(sqrt(3 ))/2 = sqrt(3 )`

so `1/(cos(30^o)+sin(60^o))=1/sqrt(3)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Putting it all together we have:

`1/sqrt(2)xx1/(sqrt(3))color(white)("d") -> color(white)("d")1/(sqrt(2)sqrt(3))`

`color(white)("dddddddddd")->color(white)("d")1/(sqrt(6))`

It is considered bad mathematical practice to have a root as or in the denominator if avoidable

Multiply by 1 and you do not change the value. However, one comes in many forms.

`color(white)("dddddddddd")->color(white)("d")1/(sqrt(6))xx1`

`color(white)("dddddddddd")->color(white)("d")1/(sqrt(6))xx(sqrt6)/(sqrt6)`

`color(white)("dddddddddd")->color(white)("d")(sqrt(6))/6`