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Value of cos 18 degree and sin 18 degree |
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Answer» Answer: Step-by-step explanation: Let, A = 18° Therefore, 5A = 90° ⇒ 2A + 3A = 90˚ ⇒ 2A = 90˚ - 3A Taking sine on both sides, we get sin 2A = sin (90˚ - 3A) = cos 3A ⇒ 2 sin A cos A = 4 cos3 A - 3 cos A ⇒ 2 sin A cos A - 4 cos3 A + 3 cos A = 0 ⇒ cos A (2 sin A - 4 COS2 A + 3) = 0 Dividing both sides by cos A = cos 18˚ ≠ 0, we get ⇒ 2 sin A - 4 (1 - sin2 A) + 3 = 0 ⇒ 4 sin2 A + 2 sin A - 1 = 0, which is a quadratic in sin A Therefore, sin A = −2±−4(4)(−1)√2(4) ⇒ sin A = −2±4+16√8 ⇒ sin A = −2±25√8 ⇒ sin A = −1±5√4 Now sin 18° is positive, as 18° lies in FIRST quadrant. Therefore, sin 18° = sin A = √5−14 Now cos 18° = √(1 - sin2 18°), [Taking positive value, cos 18° > 0] ⇒ cos 18° = 1−(5√−14)2−−−−−−−−−−√ ⇒ cos 18° = 16−(5+1−25√)16−−−−−−−−−−√ ⇒ cos 18° = 10+25√16−−−−−−√ Therefore, cos 18° = 10+25√√4 |
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