Questions: Find the exact value of the expression. tan(19π/4)

Solution Steps

To find the exact value of \(\tan \left(\frac{19 \pi}{4}\right)\), we can use the periodicity property of the tangent function, which has a period of \(\pi\). This means \(\tan(x) = \tan(x + n\pi)\) for any integer \(n\). We can reduce the angle \(\frac{19 \pi}{4}\) to an equivalent angle within the interval \([0, \pi)\) and then find the tangent of that angle.

The given angle is: \[ \frac{19\pi}{4} \]

Using the periodicity of the tangent function, we reduce the angle to an equivalent angle within the interval \([0, \pi)\): \[ \frac{19\pi}{4} \mod \pi = 2.3562 \]

Calculate the tangent of the reduced angle: \[ \tan(2.3562) = -1.0000 \]

Final Answer