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Chapter 4: Problem 70

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

Short Answer

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

Step by step solution

01

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 \(\frac{7 \pi}{4}\).
02

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, \(\cot \frac{7 \pi}{4} = \frac{cos(315)}{sin(315)}\).
03

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.

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