SOLUTION: Find the Exact value (No Decimals) of sin(105 degrees) using the sum or difference formula. I know the Sum and Difference formulas for sin : sin(alpha + beta)= sin alpha cos bet

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Question 869031: Find the Exact value (No Decimals) of sin(105 degrees) using the sum or difference formula. I know the Sum and Difference formulas for sin :
sin(alpha + beta)= sin alpha cos beta + cos alpha sin beta
sin(alpha - beta)= sin alpha cos beta - cos alpha sin beta
But what do i plug in and how does it work out?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your problem is:

Find the exact value (no decimals) of sine of 105 degrees using the sum or difference formula.

sin(105) is equal to sin(60 + 45).

since you can find the exact sine and cosine values of 60 and 45 degrees, you can use this the sum formula to find the exact value of sine of 105.

yuour sum and fiference formulas are:

sin(alpha + beta)= sin alpha cos beta + cos alpha sin beta
sin(alpha - beta)= sin alpha cos beta - cos alpha sin beta

your sum and difference formulas are correct.

you only need to use the sum formula.

that formula is:

sin (a + b) = sin(a) * cos(b) + cos(a) * sin(b) now becomes:

let a be equal to 60 degrees.
let b be equal to 45 degrees.

sin (a + b) = sin(a) * cos(b) + cos(a) * sin(b) now becomes:

sin (60 + 45) = sin(60) * cos(45) + cos(60) * sin(45).

sin (60) is equal to sqrt(3)/2

cos(60) is equal to 1/2.

sin(45) is equal to sqrt(2)/2

cos(45) is equal to sqrt(2)/2

you're good to go.

just replace sin(60) with sqrt(3)/2 and replace cos(60) with 1/2 and replace sin(45) with sqrt(2)/2 and replace cos(45) with sqrt(2)/2 and solve.

your formula of:

sin (60 + 45) = sin(60) * cos(45) + cos(60) * sin(45) becomes:

sin (105) = sqrt(3)/2 * sqrt(2)/2 + (1/2) * sqrt(2)/2.

this becomes:

sin(105) = sqrt(6)/4 + sqrt(2)/4 which becomes:

sin(105) = (sqrt(6) + sqrt(2)) / 4.

you can use your calculator to confirm this solution is good.

the calculator says that sin(105) is equal to .965925826

the calculator says that (sqrt(6) + sqrt(2)) / 4 is equal to .965925826.

since both answers are the same, you did good.