Question

let θ be an angle in standard position. name the quadrant in which θ lies. sinθ<0, cotθ<0 the angle θ lies in which quadrant? iii iv i ii

Explanation:

Step1: Analyze \(\sin\theta < 0\)

Sine of an angle is the ratio of the opposite side to the hypotenuse in a right - triangle, or in terms of the unit circle, \(y\) - coordinate of the point on the unit circle corresponding to angle \(\theta\). \(\sin\theta<0\) when the \(y\) - coordinate of the point \((x,y)\) on the unit circle is negative. This occurs in Quadrants III and IV (since in Quadrant I, \(y>0\); Quadrant II, \(y > 0\); Quadrant III, \(y<0\); Quadrant IV, \(y < 0\)).

Step2: Analyze \(\cot\theta<0\)

We know that \(\cot\theta=\frac{\cos\theta}{\sin\theta}\). For \(\cot\theta < 0\), the numerator and the denominator must have opposite signs (because a fraction is negative when the numerator and denominator have opposite signs). We already know that \(\sin\theta<0\) from Step 1, so for \(\frac{\cos\theta}{\sin\theta}<0\), \(\cos\theta\) must be positive (since \(\sin\theta\) is negative and negative divided by positive is negative). \(\cos\theta\) is positive when the \(x\) - coordinate of the point \((x,y)\) on the unit circle is positive. This occurs in Quadrants I and IV (since in Quadrant I, \(x>0\); Quadrant II, \(x < 0\); Quadrant III, \(x<0\); Quadrant IV, \(x>0\)).

Step3: Find the common quadrant

From Step 1, \(\theta\) is in Quadrants III or IV. From Step 2, \(\theta\) is in Quadrants I or IV. The common quadrant between Quadrants III, IV and Quadrants I, IV is Quadrant IV.

Answer:

IV