إذا كان ظا(و) = 4 حيث (و) زاوية منعكسة، فأوجد النسب المثلثية الأساسية والثانوية للزاوية و.

Steps to Solve

1. Understanding the given information

We're given that $\tan(\theta) = 4$, where $\theta$ is a reflex angle. Reflex angles are angles between $180^\circ$ and $360^\circ$. We need to determine the possible values of the other trigonometric functions. The question appears to want us to identify other possible trigonometric values, given the value of the tangent

2. Determining the quadrant

Since $\tan(\theta) = 4 > 0$, the angle $\theta$ must lie in either the first or third quadrant. Since we're given that $\theta$ is a reflex angle, it must lie in the third quadrant ($180^\circ < \theta < 270^\circ$).

3. Finding $\sec(\theta)$

We know the trigonometric identity $\sec^2(\theta) = 1 + \tan^2(\theta)$. Substituting $\tan(\theta) = 4$, we get $\sec^2(\theta) = 1 + 4^2 = 1 + 16 = 17$. Therefore, $\sec(\theta) = \pm\sqrt{17}$. Since $\theta$ is in the third quadrant, $\cos(\theta)$ is negative, and thus $\sec(\theta)$ is also negative. Therefore, $\sec(\theta) = -\sqrt{17}$.

4. Finding $\cos(\theta)$ Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we have $\cos(\theta) = \frac{1}{\sec(\theta)} = \frac{1}{-\sqrt{17}} = -\frac{1}{\sqrt{17}}$.

5. Finding $\sin(\theta)$ We know that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Thus, $\sin(\theta) = \tan(\theta) \cdot \cos(\theta) = 4 \cdot \left(-\frac{1}{\sqrt{17}}\right) = -\frac{4}{\sqrt{17}}$.

6. Finding $\csc(\theta)$ Since $\csc(\theta) = \frac{1}{\sin(\theta)}$, we have $\csc(\theta) = \frac{1}{-\frac{4}{\sqrt{17}}} = -\frac{\sqrt{17}}{4}$.

7. Finding $\cot(\theta)$ Since $\cot(\theta) = \frac{1}{\tan(\theta)}$, we have $\cot(\theta) = \frac{1}{4}$.

8. Matching with the options The options are written in Arabic, but translate as follows:

9. جا ه → $\sin \theta$

10. جتا ه → $\cos \theta $

11. ظتا ه → $\cot \theta$

12. قاطع ه → $\sec \theta$

13. قتا ه → $\csc \theta$

Looking at the options again we see the relevant option to be "قاطع هـ". So we are looking to see if any of the options match our results. $\sec(\theta) = -\sqrt{17}$. We have an answer choice of "قاطع هـ" or "$\sec \theta$". Thus, the correct answer is $\sec \theta$.