Understanding how trigonometric ratios change in different quadrants is really important for learning trigonometry in Pre-Calculus. Let’s break it down step by step.

The coordinate plane is divided into four quadrants:

1. Quadrant I: Both $x$ and $y$ coordinates are positive.
2. Quadrant II: $x$ is negative, while $y$ is positive.
3. Quadrant III: Both $x$ and $y$ are negative.
4. Quadrant IV: $x$ is positive, but $y$ is negative.

Now, let’s look at the trigonometric ratios—sine ($\sin$), cosine ($\cos$), and tangent ($\tan$)—and see how they change in each quadrant.

Quadrant I:

• In this quadrant, all trigonometric functions are positive:
  • $\sin \theta > 0$
  • $\cos \theta > 0$
  • $\tan \theta > 0$

This area is pretty straightforward, and it’s where most students feel comfortable. The angle you use is simply the reference angle.

Quadrant II:

• Here, sine stays positive while cosine and tangent turn negative:
  • $\sin \theta > 0$
  • $\cos \theta < 0$
  • $\tan \theta < 0$

This is common on the unit circle. Remember, sine is connected to the $y$ values, which are still above the x-axis in this quadrant.

Quadrant III:

• Everything flips! In this quadrant, both sine and cosine become negative:
  • $\sin \theta < 0$
  • $\cos \theta < 0$
  • $\tan \theta > 0$

This can be tricky, because the tangent is positive. This happens because tangent is the ratio of sine to cosine.

Quadrant IV:

• Finally, in this quadrant, sine is negative, but cosine is positive:
  • $\sin \theta < 0$
  • $\cos \theta > 0$
  • $\tan \theta < 0$

This shows that tangent is negative because it’s based on a negative sine and a positive cosine.

Summary of Signs:

Here’s a quick summary of the signs for the trigonometric functions in each quadrant:

• Quadrant I: All positive ($\sin, \cos, \tan$)
• Quadrant II: $\sin > 0$, $\cos < 0$, $\tan < 0$
• Quadrant III: $\sin < 0$, $\cos < 0$, $\tan > 0$
• Quadrant IV: $\sin < 0$, $\cos > 0$, $\tan < 0$

Knowing how these signs change is important when solving problems and finding angles. It might take some practice to remember this, but once you get the hang of it, it makes trigonometry a lot easier! Visuals like the unit circle can really help you see these changes. Happy studying!