Given: f(x)=2 cos x and g(x)=sin x. Which of these expressions is equivalent to (fxg)(π/16)?
a) cos π/8
b) sin π/8
c) cos π/4
d) sin π/4
Please explain the answer, thank you.
since (fxg)(x) = 2cosxsinx
and 2sinxcosx = sin 2x
we would be finding sin(2(π/16)) = sin π/8
To find the expression equivalent to (f x g)(π/16), we need to evaluate (f x g)(π/16).
First, let's find (f x g)(x). The notation (f x g)(x) represents the composition of functions f and g, where the output of g is used as the input for f.
If f(x) = 2 cos x and g(x) = sin x, then (f x g)(x) = f(g(x)) = 2 cos(sin x).
Now, let's evaluate (f x g)(π/16):
(f x g)(π/16) = 2 cos(sin (π/16)).
To determine the equivalent expression, we need to simplify 2 cos(sin (π/16)).
sin (π/16) is not one of the standard angles whose sine values are known. So, we cannot directly simplify the expression.
To find the equivalent expression, we need to use the angle addition formula for cosine:
cos(a + b) = cos a cos b - sin a sin b.
In this case, let a = 0 and b = sin (π/16).
cos(0 + sin (π/16)) = cos 0 cos(sin (π/16)) - sin 0 sin(sin (π/16)).
cos(0) = 1 and sin(0) = 0, so the expression simplifies to:
cos(sin (π/16)).
Therefore, the equivalent expression to (f x g)(π/16) is cos(sin (π/16)).
Looking at the options given, we can see that the correct choice is:
a) cos π/8.
This matches the simplified expression cos(sin (π/16)), which is equivalent to (f x g)(π/16).
To find the expression equivalent to (f x g)(π/16), we need to find the product of f(x) and g(x) evaluated at x = π/16.
First, let's calculate f(π/16) and g(π/16):
f(π/16) = 2 cos(π/16)
g(π/16) = sin(π/16)
Now, we can find the product (f x g)(π/16) by multiplying these two values together:
(f x g)(π/16) = f(π/16) * g(π/16) = (2 cos(π/16)) * (sin(π/16))
To simplify this expression, we can use the double-angle identity for sine:
sin(2θ) = 2sin(θ)cos(θ)
In our case, θ = π/16:
sin(π/8) = 2sin(π/16)cos(π/16)
Now, we can substitute this expression back into (f x g)(π/16):
(f x g)(π/16) = (2 cos(π/16)) * (sin(π/16)) = (2 cos(π/16)) * (sin(2π/16))
Since sin(2π/16) = sin(π/8), we have:
(f x g)(π/16) = (2 cos(π/16)) * (sin(2π/16)) = (2 cos(π/16)) * (sin(π/8))
Looking at the answer choices, we can see that the expression (fxg)(π/16) is equivalent to sin(π/8).
Therefore, the correct answer is b) sin π/8.