Questions: If f(x)=cot(x) and f(a)=7, find the exact value of the following. (a) f(-a) (b) f(a)+f(a+π)+f(a+2π) (a) f(-a)=-7 (b) f(a)+f(a+π)+f(a+2π)=

Transcript text: If $f(x)=\cot (x)$ and $f(a)=7$, find the exact value of the following. (a) $f(-a)$ (b) $f(a)+f(a+\pi)+f(a+2 \pi)$ (a) $f(-a)=-7$ (b) $f(a)+f(a+\pi)+f(a+2 \pi)=$

Solution

Solution Steps

To solve the given problems, we need to use properties of the cotangent function and its periodicity.

(a) Since $ f(x) = \cot(x) $, and knowing that $\cot(-x) = -\cot(x)$, we can find $ f(-a) $ by using the fact that $ f(a) = 7 $.

(b) The cotangent function has a period of $\pi$, meaning $ \cot(x + \pi) = -\cot(x) $. Therefore, we can use this property to find $ f(a) + f(a + \pi) + f(a + 2\pi) $.

Step 1: Calculate $ f(-a) $

Given that $ f(a) = 7 $ and using the property of the cotangent function, we have: \[ f(-a) = -f(a) = -7 \]

Step 2: Calculate $ f(a) + f(a + \pi) + f(a + 2\pi) $

Using the periodicity of the cotangent function: