<h3>Answer: D) none of the equations are identities</h3>
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Work Shown:
I'm going to use x in place of theta.
let
f(x) = sin(x+2pi/4)
g(x) = sqrt(2)*sin(x)*cos(x)
Plug in x = 0
f(x) = sin(x+2pi/4)
f(0) = sin(0+2pi/4)
f(0) = sin(2pi/4)
f(0) = sin(pi/2)
f(0) = 1
and
g(x) = sqrt(2)*sin(x)*cos(x)
g(0) = sqrt(2)*sin(0)*cos(0)
g(0) = sqrt(2)*0*1
g(0) = 0
We see that f(0) = g(0) is not a true equation
Therefore, f(x) = g(x) is not true for all values of x, making it not be an identity. We can use a graph to compare f(x) and g(x) to see that they do not line up.
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Let
h(x) = cos(x-pi/3)
i(x) = (3/sqrt(2))*(cos(x) - sin(x))
Plug in x = 0
h(x) = cos(x-pi/3)
h(0) = cos(0-pi/3)
h(0) = 0.5
and
i(x) = (3/sqrt(2))*(cos(x) - sin(x))
i(0) = (3/sqrt(2))*(cos(0) - sin(0))
i(0) = 2.12
Like with equation 1, we do not have an identity since h(0) does not equal i(0). We only need one counter example since an identity states that h(x) = i(x) for all values of x in the domain.
