Questions: Determine if the given value of (x) is a solution to the given equation. [ 3 sec ^2(x)=4 ; x=pi ]

Solution Steps

To determine if the given value of \( x \) is a solution to the equation \( 3 \sec^2(x) = 4 \), we need to substitute \( x = \pi \) into the equation and check if both sides are equal. The secant function, \(\sec(x)\), is the reciprocal of the cosine function, so we will calculate \(\sec(\pi)\) and then evaluate \( 3 \sec^2(\pi) \) to see if it equals 4.

We start by substituting \( x = \pi \) into the equation \( 3 \sec^2(x) = 4 \).

The secant function is defined as: \[ \sec(x) = \frac{1}{\cos(x)} \] Calculating \( \sec(\pi) \): \[ \sec(\pi) = \frac{1}{\cos(\pi)} = \frac{1}{-1} = -1 \]

Now we compute \( 3 \sec^2(\pi) \): \[ \sec^2(\pi) = (-1)^2 = 1 \] Thus, \[ 3 \sec^2(\pi) = 3 \cdot 1 = 3 \]

We need to check if \( 3 \sec^2(\pi) \) equals 4: \[ 3 \neq 4 \]

Final Answer