Question

question 2
determine which quadrant each angle will terminate. type 1, 2, 3 or 4 for the corresponding quadrant.
-100°
170°
-\frac{7\pi}{6}
400°
5 radians
290°
-225°
\frac{5\pi}{7}

Explanation:

Step1: Recall quadrant - angle ranges

Quadrant 1: $0^{\circ}<\theta < 90^{\circ}$ or $0 < \theta<\frac{\pi}{2}$ radians; Quadrant 2: $90^{\circ}<\theta < 180^{\circ}$ or $\frac{\pi}{2}<\theta<\pi$ radians; Quadrant 3: $180^{\circ}<\theta < 270^{\circ}$ or $\pi<\theta<\frac{3\pi}{2}$ radians; Quadrant 4: $270^{\circ}<\theta < 360^{\circ}$ or $\frac{3\pi}{2}<\theta<2\pi$ radians. For negative angles, we rotate clock - wise.

Step2: Analyze $- 100^{\circ}$

Since $-90^{\circ}>-100^{\circ}>-180^{\circ}$, it terminates in Quadrant 3.

Step3: Analyze $170^{\circ}$

Since $90^{\circ}<170^{\circ}<180^{\circ}$, it terminates in Quadrant 2.

Step4: Analyze $-\frac{7\pi}{6}$

First, convert to degrees: $-\frac{7\pi}{6}\times\frac{180^{\circ}}{\pi}=-210^{\circ}$. Since $-180^{\circ}>-210^{\circ}>-270^{\circ}$, it terminates in Quadrant 2.

Step5: Analyze $400^{\circ}$

$400^{\circ}=360^{\circ} + 40^{\circ}$. Since $0^{\circ}<40^{\circ}<90^{\circ}$, it terminates in Quadrant 1.

Step6: Analyze 5 radians

Convert to degrees: $5\times\frac{180^{\circ}}{\pi}\approx286.5^{\circ}$. Since $270^{\circ}<286.5^{\circ}<360^{\circ}$, it terminates in Quadrant 4.

Step7: Analyze $290^{\circ}$

Since $270^{\circ}<290^{\circ}<360^{\circ}$, it terminates in Quadrant 4.

Step8: Analyze $-225^{\circ}$

Since $-180^{\circ}>-225^{\circ}>-270^{\circ}$, it terminates in Quadrant 2.

Step9: Analyze $\frac{5\pi}{7}$

Convert to degrees: $\frac{5\pi}{7}\times\frac{180^{\circ}}{\pi}\approx128.6^{\circ}$. Since $90^{\circ}<128.6^{\circ}<180^{\circ}$, it terminates in Quadrant 2.

Answer:

3
2
2
1
4
4
2
2