Math / Trigonometry

QuestionDetermine the exact value of $\cot \frac{5 \pi}{3}$ .

Studdy Solution

STEP 1

What is this asking? We need to find the exact value of cotangent of $\frac{5\pi}{3}$ radians, which means we want a nice, clean fraction, not a decimal approximation! Watch out! Remember cotangent is the reciprocal of tangent, not cosine!
Also, be careful with the signs in the different quadrants.

STEP 2

1. Relate Cotangent to Tangent
2. Find the Reference Angle
3. Evaluate the Tangent
4. Flip for Cotangent

STEP 3

Alright, let's start by remembering that cotangent is the reciprocal of tangent.
So, we can write: $\cot \frac{5\pi}{3} = \frac{1}{\tan \frac{5\pi}{3}}$ This means if we find $\tan \frac{5\pi}{3}$ , we just flip it over to get our answer!

STEP 4

Now, $\frac{5\pi}{3}$ is in the fourth quadrant.
We know this because $\frac{5\pi}{3}$ is between $\frac{3\pi}{2}$ and $2\pi$ .

STEP 5

To find the reference angle, let's think about how far $\frac{5\pi}{3}$ is from the x-axis.
In the fourth quadrant, we subtract our angle from $2\pi$ : $2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}$ So, our reference angle is $\frac{\pi}{3}$ !

STEP 6

We know that $\tan \frac{\pi}{3} = \sqrt{3}$ .
But remember, we're in the fourth quadrant, where tangent is negative.
So: $\tan \frac{5\pi}{3} = -\sqrt{3}$

STEP 7

Almost there!
Now we just need to take the reciprocal of our tangent value to find the cotangent: $\cot \frac{5\pi}{3} = \frac{1}{\tan \frac{5\pi}{3}} = \frac{1}{-\sqrt{3}}$

STEP 8

Let's rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$ : $\frac{1}{-\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{-3} = -\frac{\sqrt{3}}{3}$ And there we have it!

STEP 9

The exact value of $\cot \frac{5\pi}{3}$ is $-\frac{\sqrt{3}}{3}$ .

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