If \(\theta\) lies in the third quadrant and its reference angle is \(35^\circ\), what is \(\theta\)?यदि \(\theta\) तीसरे चतुर्थांश में है और उसका संदर्भ कोण \(35^\circ\) है, तो \(\theta\) का मान क्या होगा?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \(215^\circ\)\(215^\circ\)
The direct answer is B, \\(215^\circ\\). A reference angle is the small positive angle between the terminal arm and the nearest x-axis. In the third quadrant, an angle lies between \\(180^\circ\\) and \\(270^\circ\\), and the reference angle is measured after \\(180^\circ\\). Therefore, \\(\theta=180^\circ+35^\circ=215^\circ\\). Option A, \\(145^\circ\\), is \\(180^\circ-35^\circ\\), which belongs to the second quadrant. Option B is correct because 215° is in the third quadrant and has reference angle 35°. Option C, \\(325^\circ\\), is \\(360^\circ-35^\circ\\), so it belongs to the fourth quadrant. Option D, \\(305^\circ\\), also lies in the fourth quadrant and does not have the required reference angle. Exam cue: for a third-quadrant angle use \\(180^\circ+\alpha\\), not subtraction.सीधा उत्तर B, \\(215^\circ\\) है। Reference angle वह छोटा positive angle है जो terminal arm और निकटतम x-axis के बीच बनता है। Third quadrant में कोण \\(180^\circ\\) और \\(270^\circ\\) के बीच होता है, इसलिए reference angle को \\(180^\circ\\) के बाद जोड़ा जाता है। अतः \\(\theta=180^\circ+35^\circ=215^\circ\\)। A, \\(145^\circ\\), \\(180^\circ-35^\circ\\) है और second quadrant में आता है। B सही है क्योंकि 215° third quadrant में है और उसका reference angle 35° है। C, \\(325^\circ\\), \\(360^\circ-35^\circ\\) है और fourth quadrant में आता है। D, \\(305^\circ\\), भी fourth quadrant में है और इसका reference angle 35° नहीं है। परीक्षा संकेत: third quadrant के लिए \\(180^\circ+\alpha\\) लगाएँ, घटाएँ नहीं।
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