find the exact value cos (-5 pi/ 12)
( - 5 pi / 12 ) rad = - 75 °
cos ( - x ) = cos x
cos ( - 5 pi / 12 ) rad = cos ( 75 ° )
75 ° = 45 ° + 30 °
cos ( A + B ) = cos ( A ) * cos ( B ) - sin ( A ) * Sin ( B )
cos ( 75 ° ) = cos ( 45 ° + 30 ° ) = cos ( 45° ) * cos ( 30 ° ) - sin ( 45 ° ) * sin ( 30 ° )
Calculate this.
Exact value :
cos ( - 5 pi / 12 ) = ( sqrt 3 - 1 ) / ( 2 * sqrt 2 )
Suppose sin x =4/5 , 0<x<pi/2 and cos y = -1/2, pi/2 <y< pi
evaluate
cos (x-y)
To find the exact value of cos(-5π/12), we can use the cosine of a difference formula.
The cosine of a difference formula states that cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
In this case, we want to find cos(-5π/12), so A = 0 and B = 5π/12.
Using the formula, we have:
cos(-5π/12) = cos(0 - 5π/12) = cos(0)cos(5π/12) + sin(0)sin(5π/12)
Since cos(0) = 1 and sin(0) = 0, we can simplify it to:
cos(-5π/12) = cos(5π/12)
Now, let's find cos(5π/12). We can use the half-angle formula for cosine:
cos(θ/2) = ±√((1 + cosθ)/2)
In this case, θ = 5π/6, so we have:
cos(5π/12) = cos((5π/6) / 2) = cos(5π/12)
Substituting into the half-angle formula, we get:
cos(5π/12) = ±√((1 + cos(5π/6)) / 2)
Now, let's find cos(5π/6):
cos(5π/6) = cos(π - π/6) = cos(π)cos(π/6) + sin(π)sin(π/6)
Since cos(π) = -1 and sin(π) = 0, we can simplify it to:
cos(5π/6) = -cos(π/6)
Using the half-angle formula again, we get:
cos(π/6) = ±√((1 + cos(π/3)) / 2)
Now, let's find cos(π/3):
cos(π/3) = 1/2
Substituting back into the previous equation:
cos(π/6) = ±√((1 + 1/2) / 2) = ±√(3/4) = ±√3/2
Finally, substituting back into the first equation:
cos(5π/12) = ±√((1 + (-√3/2)) / 2) = ±√((2 - √3)/4) = ±√(2 - √3)/2
Therefore, the exact value of cos(-5π/12) is ±√(2 - √3)/2.