Which of the following values for theta is a counterexample to the claim that cos(pi-theta)=cos (theta) is an identity?
a)
cos(pi-theta)=cos (theta) is an identity
b)
pi/2
c)
3pi/2
d)
pi
AAAaannndd the bot gets it wrong yet again!
cos(π - 3π/2) = cos(-π/2) = 0
cos 3π/2 = 0
For π,
cos(π-π) = cos(0) = 1
cos(π) = -1
I apologize for the mistake in my previous response. You are correct. The correct answer is d) pi. Thank you for correcting me.
To find if any of the given values of theta form a counterexample to the claim that cos(pi-theta)=cos (theta) is an identity, we need to substitute each value into the equation and check if it holds true.
Let's start by substituting option b) theta = pi/2:
cos(pi - pi/2) = cos(pi/2)
Simplifying the left side:
cos(2pi/2 - pi/2) = cos(pi/2)
cos(pi/2) = cos(pi/2)
The equation holds true for this value of theta.
Next, let's substitute option c) theta = 3pi/2:
cos(pi - 3pi/2) = cos(3pi/2)
Simplifying the left side:
cos(2pi/2 - 3pi/2) = cos(3pi/2)
cos(-pi/2) = cos(3pi/2)
We know that the cosine function is an even function, which means cos(-x) = cos(x). Therefore:
cos(-pi/2) = cos(3pi/2)
cos(pi/2) = cos(3pi/2)
The equation holds true for this value of theta as well.
Lastly, let's substitute option d) theta = pi:
cos(pi - pi) = cos(pi)
Simplifying both sides:
cos(0) = cos(pi)
1 = -1
Here we can clearly see that the equation doesn't hold true for this value of theta.
Therefore, the correct answer is d) theta = pi, as it forms a counterexample to the claim that cos(pi-theta)=cos (theta) is an identity.