Question

Name the quadrant in which the angle \(\theta\) lies. $$ \sin \theta<0, \quad \cot \theta>0 $$

Short answer

Third Quadrant

Step-by-step solution

Identify the Sine Condition

Given that \( \sin \theta < 0\), recall that sine is negative in the third and fourth quadrants.

Identify the Cotangent Condition

Given that \( \cot \theta > 0\), remember that cotangent is positive when both sine and cosine are either both positive or both negative. Since sine is negative, cosine must also be negative, which occurs in the third quadrant.

Combine the Conditions

The conditions \( \sin \theta < 0\) and \( \cot \theta > 0\) occur together in the third quadrant, where both sine and cosine are negative.

Key concepts

negative sine

When we talk about the sine function, it helps to understand that sine values come from the y-coordinates on the unit circle. If \(\text{sin} \theta < 0\), this means that the y-coordinate is negative. In simpler terms, the terminal side of the angle \(\theta\) lies below the x-axis on the unit circle.
This gives an indication that \(\theta\) must be in either the third or fourth quadrant because these are the quadrants where the sine values are negative.
Remember:

• First and second quadrants have positive sine values.
• Third and fourth quadrants have negative sine values.

Thus, knowing that \(\text{sin} \theta < 0\) helps narrow down our options to only the third and fourth quadrants.

positive cotangent

Cotangent is the reciprocal of the tangent function. Mathematically, \(\cot\)=\frac{1}{tan}\ or \(\cot\theta = \frac{\cos\theta}{\sin\theta}\). So, positive cotangent means that the product of cosine and sine, or the ratio of cosine to sine, is positive.
For cotangent to be positive, two cases arise:

• Both sine and cosine are positive.
• Both sine and cosine are negative.

Since we've already identified that \(\text{sin} \theta < 0\), this means sine is negative. Consequently, for cotangent to remain positive, cosine must also be negative. This simultaneously narrows down our quadrant selection further.
In which quadrant is both sine and cosine negative? It's the third quadrant!

third quadrant

The third quadrant of the unit circle is defined where both x and y coordinates are negative. This means both cosine and sine values are negative in this quadrant. Using the principles we discussed:

• Sine negativity narrows our options to the third and fourth quadrants.
• Positive cotangent indicates both sine and cosine must be negative.

By combining these two, we can definitively say that the angle \(\theta\) is in the third quadrant.
Thus, the final answer to the problem is that angle \(\theta\) lies in the Third Quadrant.