Question

which of the following angles does not terminate in quadrant 3? choose all that apply. -160°; 7π/5; 6 radians; 14π/9; -405°; 4 radians. question 3 which statement is true given that θ is an angle in quadrant 3? cscθ and cotθ are both negative; cscθ and cosθ are both positive; sinθ and secθ are both negative; cscθ and tanθ are both positive

Explanation:

Step1: Recall quadrant - angle ranges

Angles in Quadrant 3 satisfy \(180^{\circ}+ 360^{\circ}n<\theta<270^{\circ}+360^{\circ}n\) (in degrees) or \(\pi + 2k\pi<\theta<\frac{3\pi}{2}+2k\pi\) (in radians), where \(n,k\in\mathbb{Z}\).

Step2: Analyze \(-160^{\circ}\)

\(-160^{\circ}=-360^{\circ}+200^{\circ}\), and \(180^{\circ}<200^{\circ}<270^{\circ}\), so it is in Quadrant 3.

Step3: Analyze \(\frac{7\pi}{5}\)

\(\pi=\frac{5\pi}{5}<\frac{7\pi}{5}<\frac{3\pi}{2}=\frac{7.5\pi}{5}\), so it is in Quadrant 3.

Step4: Analyze 6 radians

\(2\pi\approx6.28\), \(\frac{3\pi}{2}\approx4.71\), \( \frac{3\pi}{2}<6<2\pi\), so it is in Quadrant 4.

Step5: Analyze \(\frac{14\pi}{9}\)

\(\frac{14\pi}{9}\approx1.56\pi\), \(\pi<1.56\pi<\frac{3\pi}{2}\), so it is in Quadrant 3.

Step6: Analyze \(-405^{\circ}\)

\(-405^{\circ}=-360^{\circ}- 45^{\circ}\), it is in Quadrant 4.

Step7: Analyze 4 radians

\(\pi\approx3.14\), \(\frac{3\pi}{2}\approx4.71\), \(\pi<4<\frac{3\pi}{2}\), so it is in Quadrant 3.

Step8: Recall trig - function signs in Quadrant 3

In Quadrant 3, \(\sin\theta<0\), \(\cos\theta<0\), \(\tan\theta>0\), \(\cot\theta>0\), \(\sec\theta=\frac{1}{\cos\theta}<0\), \(\csc\theta=\frac{1}{\sin\theta}<0\).

Answer:

• First question: 6 radians, \(-405^{\circ}\)
• Second question: \(\sin\theta\) and \(\sec\theta\) are both negative