1. **Problem:** Given the diagram of △ABC with angles B = 50°, A = 80°, and side BC = 20 units, find the value of sec A. **Step 1:** Recall that \(\sec A = \frac{1}{\cos A}\). **Step 2:** Calculate \(\cos 80^\circ\). $$\cos 80^\circ \approx 0.1736$$ **Step 3:** Calculate \(\sec 80^\circ = \frac{1}{0.1736} \approx 5.76$$ **Step 4:** Check the options: a) 12/13 \(\approx 0.923\), b) 13/12 \(\approx 1.083\), c) 12/5 \(= 2.4\), d) 13/5 \(= 2.6\). None match 5.76 exactly, but since the triangle sides are given, let's verify if the side lengths correspond to any of these ratios. **Step 5:** Using Law of Cosines to find side AC or AB is complex here; however, the closest option to \(\sec 80^\circ\) is none. Possibly a typo or the question expects \(\sec A = \frac{hypotenuse}{adjacent} = \frac{13}{5} = 2.6\) if side lengths are 5 and 13. **Answer:** d) 13/5 2. **Problem:** Given \(\cos \theta = -0.2489\), find the quadrants where the principal angle lies. **Step 1:** Cosine is negative in Quadrants II and III. **Answer:** c) Q2 and Q3 3. **Problem:** Determine the number of triangles that can be drawn with \(a=4.7\) cm, \(b=8.3\) cm, and \(\angle A=37^\circ\). **Step 1:** Use the Law of Sines to find possible \(\angle B\). $$\sin B = \frac{b \sin A}{a} = \frac{8.3 \times \sin 37^\circ}{4.7}$$ Calculate \(\sin 37^\circ \approx 0.6018\). $$\sin B = \frac{8.3 \times 0.6018}{4.7} \approx \frac{4.995}{4.7} \approx 1.063$$ Since \(\sin B > 1\), no triangle can be formed. **Answer:** a) 0 4. **Problem:** Which pair of angles are not co-terminal? **Step 1:** Two angles are co-terminal if their difference is a multiple of 360°. Check each pair: - a) 23° and 383°: 383 - 23 = 360 → co-terminal - b) -105° and -465°: -465 - (-105) = -360 → co-terminal - c) -123° and 237°: 237 - (-123) = 360 → co-terminal - d) -19° and 390°: 390 - (-19) = 409 → not a multiple of 360 **Answer:** d) –19°, 390° 5. **Problem:** Given △ABC, determine the most appropriate trigonometric tool to find side a. **Step 1:** If two sides and an angle opposite one side are known, use Sine Law. **Answer:** d) Sine Law 6. **Problem:** Which expression is equivalent to \(\frac{\sin \theta}{\tan \theta}\)? **Step 1:** Recall \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). **Step 2:** Substitute: $$\frac{\sin \theta}{\tan \theta} = \frac{\sin \theta}{\frac{\sin \theta}{\cos \theta}} = \cos \theta$$ **Answer:** a) cos \(\theta\)