Question

Determine the exact value of (sin 45 °) (cos 45 ) + (sin 30°) (sin 60°)

(Don't leave radicals in the denominator)

Answer 1

`(1)/(2)+ (√(3) )/(4)`

Explanation:

Refer to the triangle diagrams I attached. Use SOH CAH TOA

`sin(45)*cos(45)+sin (30)* sin (60)`

The exact value of sin(45) is
`(√(2) )/(2)`

`(√(2) )/(2)*cos(45)+sin (30)* sin (60)`

The exact value of cos(45) is
`(√(2) )/(2)`

`(√(2) )/(2)*(√(2) )/(2)+sin (30)* sin (60)`

Multiply
`(√(2) )/(2)*(√(2) )/(2)`.

`(√(2)^(2) )/(4)+sin (30)* sin (60)`

Rewrite
`{√(2)^(2)` as 2.

`(2)/(4) +sin (30)* sin (60)`

Simplify

`(1)/(2) +sin (30)* sin (60)`

The exact value of sin(30) is
`(1)/(2)`

`(1)/(2) +(1)/(2) * sin (60)`

The exact value of sin(60) is
`(√(3) )/(2)`

`(1)/(2) +(1)/(2) * (√(3) )/(2)`

Simplify

`(1)/(2)+ (√(3) )/(4)`