Question

How do i find the exact value of `sin33cos27 + cos33sin27`? :/ without using calculator.

Answer 1

We can use the sum of two angles identity `sin(A+B)=sinAcosB+sinBcosA`

`sin33cos27+cos33sin27=sin(33+27)=sin(60)`

We know that `sin\theta=O /H` and that an equilateral triangle has three angles of `60^circ`.

If we divide the triangle (with sides of 1) in two, we get a right angle triangle of sides 1, 0.5 and `x`. `x=sqrt(1^2-0.5^2)=sqrt(3/4)=sqrt(3)/sqrt(4)=sqrt(3)/2`

`sin60=O /H=((sqrt(3)/2))/1=sqrt(3)/2`

Answer 2

`sqrt3/2`

Explanation:

`"using the "color(blue)"addition formula for sin"`

`•color(white)(x)sin(A+-B)=sinAcosB+-cosAsinB`

`sin33cos27+cos33sin27" is in this form"`

`"with "A=33" and "B=27`

`rArrsin(33+27)^@" represents the expansion"`

`=sin60^@`

`"we can obtain the "color(blue)"exact value"`

`"by considering the "90-60-30" triangle"`

`"with sides "1,sqrt3" and "2`

`rArrsin60^@=sqrt3/2`