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Chapter 4: Problem 84

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\tan \left(-\frac{11 \pi}{4}\right)$$

Short Answer

Expert verified
The value of \( \tan \left(-\frac{11\pi}{4}\right) \) is -1.

Step by step solution

01

Rewriting into positive representation

To find the reference angle, we need to bring the angle \( -\frac{11\pi}{4} \) into the range of the first quadrant by adding multiples of \( 2\pi \). Since \( \frac{11\pi}{4} = 2\pi + \frac{3\pi}{4} \) and \( 2\pi \) represents a full revolution, we can rewrite \( -\frac{11 \pi}{4} \) as \( \frac{3 \pi}{4} \) in the counter-clockwise direction.
02

Finding the reference angle

Now that we have the positive counter-part, we find the angle between \( \frac{3\pi}{4} \) and \( \pi \) (that is the closest x-axis). This is calculated by subtracting \( \frac{3\pi}{4} \) from \( \pi \) which gives us \( \frac{\pi}{4} \) which is the reference angle.
03

Calculating the required expression

Lastly, recognizing that the original angle \( -\frac{11\pi}{4} \) ends up in the second quadrant, where the tangent function is negative, we assign a negative sign to the tangent of the reference angle. So the value of \( \tan \left(-\frac{11\pi}{4}\right) \) becomes \( -\tan(\frac{\pi}{4}) \).
04

Final calculation

We know from basic trigonometry that \( \tan(\frac{\pi}{4}) = 1 \), therefore, \( \tan \left(-\frac{11\pi}{4}\right) = -1 \).

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