Question

How do you find the exact functional value sin375 using the cosine sum or difference identity?

Answer 1

`sin(375)=sin(15)=(sqrt(3)-1)/(2sqrt(2))`

Explanation:

I'd use the identity `cos(b) sin(a)+cos(a) sin(b)`, with `a=360°`, and `b=15°`, since both are known values:

• `cos(360)=cos(0)=1`
• `sin(360)=sin(0)=0`
• `cos(15)=(sqrt(3)+1)/(2sqrt(2))`
• `sin(15)=(sqrt(3)-1)/(2sqrt(2))`

Plugging this value into the equality gives

`sin(375)=sin(360+15)=cos(15)*sin(360)+sin(15)*cos(360)=cos(15)*0+sin(15)*1`

So, we just discovered in a quite unpractical way that `sin(375)=sin(15)=(sqrt(3)-1)/(2sqrt(2))`: a very much easier way to notice this would have been using the periodicity of the sine function: `sin(x)=sin(360+x)`.

Answer 2

Find `sin 375`

Ans: `sqrt(2 - sqrt3)/2`

Explanation:

sin (375) = sin (15 + 360) = sin 15. Call sin (15) = sin a
`cos (2a) = cos (30) = sqrt3/2`
Use the trig identity: `cos 2a = 1 - 2sin^2 a`
`sqrt3/2 = 1 - 2sin^2 a`
`2sin^2 a = 1 - sqrt3 = (2 - sqrt3)/2`
`sin^2 a = (2 - sqrt3)/4`
`sin (15) = sin a = +- sqrt(2 - sqrt3)/2`
Since arc (15) is in Quadrant I, its sin is positive,
`sin (15) = sqrt(2 - sqrt3)/2`

Check by calculator.
sin 15 = 0.26
`sqrt(2 - sqrt3)/2 = 0.52/2 = 0.26`. OK