Use the unit circle to evaluate the function. Tan 7pie/6
The Correct Answer and Explanation is:
To evaluate tan(7π/6) using the unit circle:
✅ Correct Answer:
\tan\left(\frac{7\pi}{6}\right) = \frac{1}{\sqrt{3}} \cdot (-1)^2 = \frac{1}{\sqrt{3}} \cdot 1 = \frac{1}{\sqrt{3}} \text{ with the sign } (-), \text{ so } \boxed{\tan\left(\frac{7\pi}{6}\right) = \frac{1}{\sqrt{3}} \cdot (-1) = \boxed{\frac{1}{\sqrt{3}}(-1) = \boxed{\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}}}
🧠 Step-by-Step Explanation:
- Understand the angle: 7π6 is in radians. To understand its location on the unit circle, convert it to degrees:\frac{7\pi}{6} \text{ is in radians. To understand its location on the unit circle, convert it to degrees:}67π is in radians. To understand its location on the unit circle, convert it to degrees: 7π6⋅180∘π=210∘\frac{7\pi}{6} \cdot \frac{180^\circ}{\pi} = 210^\circ67π⋅π180∘=210∘
- Find the reference angle:
The reference angle is the acute angle formed with the x-axis. Since 210° is in the third quadrant: 210∘−180∘=30∘210^\circ – 180^\circ = 30^\circ210∘−180∘=30∘ So the reference angle is 30° (or π/6 radians). - Use known values from the unit circle:
For the reference angle π/6:- sin(π/6)=12\sin(\pi/6) = \frac{1}{2}sin(π/6)=21
- cos(π/6)=32\cos(\pi/6) = \frac{\sqrt{3}}{2}cos(π/6)=23
- tan(π/6)=13=33\tan(\pi/6) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(π/6)=31=33
- Determine the sign of tangent in the third quadrant:
In the third quadrant, both sine and cosine are negative, and since tangent is sine divided by cosine, the negatives cancel out: tan(7π6)=−12−32=13=33\tan\left(\frac{7\pi}{6}\right) = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(67π)=−23−21=31=33 - Final step – correct sign:
Since both sine and cosine are negative in quadrant III, their ratio is positive. So: tan(7π6)=33\boxed{\tan\left(\frac{7\pi}{6}\right) = \frac{\sqrt{3}}{3}}tan(67π)=33
However, there is a sign mistake in the earlier step. Let’s correct:
At 210°, sine and cosine are both negative:
- sin(210∘)=−12\sin(210^\circ) = -\frac{1}{2}sin(210∘)=−21
- cos(210∘)=−32\cos(210^\circ) = -\frac{\sqrt{3}}{2}cos(210∘)=−23
So:tan(210∘)=−12−32=13=33\tan(210^\circ) = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}tan(210∘)=−23−21=31=33
Therefore, the final correct answer is:tan(7π6)=33\boxed{\tan\left(\frac{7\pi}{6}\right) = \frac{\sqrt{3}}{3}}tan(67π)=33
