Find the exact value of the expression.
cos(π/16)cos(3π/16) − sin(π/16) sin(3π/16)
To find the exact value of the expression cos(π/16)cos(3π/16) - sin(π/16)sin(3π/16), we can use the trigonometric identity for the cosine of the difference of two angles:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Let's apply this identity to the given expression:
cos(π/16)cos(3π/16) - sin(π/16)sin(3π/16)
= cos(π/16 - 3π/16)
The difference of π/16 and 3π/16 is -2π/16, which simplifies to -π/8.
So, the expression can be written as:
cos(-π/8)
Since the cosine function is an even function, cos(-θ) = cos(θ).
Therefore, the expression simplifies to:
cos(π/8)
Now, we can use the value of the cosine function at π/8.
To evaluate cos(π/8) exactly, we can use various methods such as geometric or trigonometric constructions, or use mathematical formulas and identities.
One common method is to use the half-angle identity for cosine:
cos(θ/2) = ±√((1 + cos(θ))/2)
Applying this formula to cos(π/8), we have:
cos(π/8) = cos(π/4/2) = ±√((1 + cos(π/4))/2)
Now, we know that cos(π/4) = √2/2, so we substitute this value into the formula:
cos(π/8) = ±√((1 + √2/2)/2)
To be more precise, using the positive root:
cos(π/8) = √((1 + √2/2)/2)
This is the exact value of the expression cos(π/16)cos(3π/16) - sin(π/16)sin(3π/16).