Problem 70 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\cot \frac{7 \pi}{4}$$

Step-by-Step Solution

Answer

The exact value of \( \cot \frac{7 \pi}{4} \) is 1.

Step 1: Find Reference Angle

In order to find the reference angle, we need to find the equivalent positive angle less than or equal to \(2 \pi\). The given angle is \(\frac{7 \pi}{4}\) which is larger than \(2 \pi\). To find the reference angle, we subtract multiples of \(2 \pi\) until we get an angle between 0 and \(2 \pi\). With that in mind, our reference angle will be \( \frac{7 \pi}{4} - 2 \pi = \frac{ - \pi}{4}\). However we need a positive angle, so by adding \(2 \pi\) we get the positive equivalent \(\frac{7 \pi}{4}\).

Step 2: Find Cotangent Value Using Reference Angle

Now we convert this positive angle into degrees. Multiplying it by \(\frac{180^\circ}{\pi}\), we get the angle as 315 degrees. In the unit circle, for 315 degrees, the x-coordinate translates to cos(315) and the y-coordinate to sin(315). Both of them are equal to \(\frac{\sqrt2}{2}\). By the definition of cotangent, we have \(\cot \theta = \frac{cos \theta}{sin \theta}\). Therefore, \(\cot \frac{7 \pi}{4} = \frac{cos(315)}{sin(315)}\).

Step 3: Calculate the Cotangent Value

Substitute the values of cosine and sine which were found from unit circle. \(\cot \frac{7 \pi}{4} = \frac{\sqrt2/2}{\sqrt2/2}\). The value of the cotangent simplifies to 1.