Determine the quadrant of θ if tanθ=−3√3 and cosθ=−3√2 .(1 point) Responses Quadrant II Quadrant II Quadrant I Quadrant I Quadrant III Quadrant III Quadrant IV

Answer 1

To determine the quadrant of θ given \( \tan \theta = -3\sqrt{3} \) and \( \cos \theta = -3\sqrt{2} \), we analyze the signs of the trigonometric functions.

1. Tangent (\(\tan \theta\)):

   • Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) is negative, this means that either:
     • \( \sin \theta > 0 \) and \( \cos \theta < 0 \) (which occurs in Quadrant II)
     • or \( \sin \theta < 0 \) and \( \cos \theta > 0 \) (which occurs in Quadrant IV)

2. Cosine (\(\cos \theta\)):

   • Given \( \cos \theta = -3\sqrt{2} \), this is negative, indicating that θ is in either Quadrant II or Quadrant III (since cosine is negative in these quadrants).

Combining these pieces of information:

• Since \( \tan \theta < 0 \) and \( \cos \theta < 0 \), θ must be in Quadrant II.

Thus, the quadrant of θ is Quadrant II.