rdfvjur的部落格

標題:

sum and product fromulae (math

發問:

1. http://upload.lsforum.net/users/public/w469sin1q64.png 2. cos(x+15)cos(x-15) - sinx sin (x-30) =cos165 3. cos x + cos2x+ cos3x + cos4x = 0 4. tan (30+x) tan (30-x) = 1-2cos2x 更新: 其實我不是很明白 第一題 :5x/2 = 0° or 180° or 360° or 540° or 720° or 900° 呢幾個答案360° or 540° or 720° or 900°點黎??

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1. 我接著你最後的步驟繼續做： sin(x/2) = 0 or tan(5x/2) = 0 where 0° ≤ x ≤ 360°, that is, 0° ≤ x/2 ≤ 180° and 0° ≤ 5x/2 ≤ 900° [留意一個cycle之中 sin y = 0 只有 y = 0°, 180°, 360°] [留意一個cycle之中 tan y = 0 只有 y = 0°, 180°, 360°] 所以， x/2 = 0° or 180° or 5x/2 = 0° or 180° or 360° or 540° or 720° or 900° 因此， x = 0° or 360° or 72° or 144° or 216° or 288° 〔共六個根。〕 2. cos(x+15°)cos(x-15°) - sinx sin (x-30°) = cos165° (1/2)[cos(2x)+cos(30°)] - (1/2)[cos(30°)-cos(2x-30°)] = cos165° [cos(2x)+cos(30°)] - [cos(30°)-cos(2x-30°)] = 2cos165° cos(2x) + cos(2x-30°) = 2cos165° 2 cos(2x-15°) cos(15°) = 2cos(180°-15°) 2 cos(2x-15°) cos(15°) = -2cos(15°) cos(2x-15°) = -1 2x-15° = 180° or 540° 2x = 195° or 555° x = 97.5° or 277.5° 〔共兩個根。〕 3. cos x + cos2x + cos3x + cos4x = 0 (cos x + cos4x) + (cos2x + cos3x) = 0 2 cos (5x/2) cos(-3x/2) + 2 cos (5x/2) cos(-x/2) = 0 2 cos (5x/2) [cos(-3x/2) + cos(-x/2)] = 0 2 cos (5x/2) [ 2 cos(-x) cos(-x/2) ] = 0 4 cos (5x/2) cos(x) cos(x/2) = 0 cos (5x/2) = 0 or cos(x) = 0 or cos(x/2) = 0 5x/2 = 90° or 270° or 450° or 630° or 810° or x = 90° or 270° or x/2 = 90° x = 36° or 108° or 180° or 252° or 324° or 90° or 270° 〔共七個根。〕 4. tan (30°+x) tan (30°-x) = 1 - 2cos2x [tan(30°)+tan(x)]/[1-tan(30°)tan(x)]*[tan(30°)-tan(x)]/[1+tan(30°)tan(x)] = 1 - 2cos2x [tan2(30°)-tan2(x)]/[1-tan2(30°)tan2(x)] = 1 - 2(cos2x - sin2x) [1/3-tan2(x)]/[1-tan2(x)/3] = (cos2x + sin2x) - 2(cos2x - sin2x) [1-3tan2(x)]/[3-tan2(x)] = 3sin2x - cos2x [cos2x-3sin2(x)]/[3cos2x-sin2(x)] = 3sin2x - cos2x [cos2x-3sin2(x)] = [3sin2x - cos2x][3cos2x-sin2(x)] [3sin2x - cos2x][3cos2x-sin2(x)] - [cos2x-3sin2(x)] =0 [3sin2x - cos2x][3cos2x-sin2(x)] + [3sin2x - cos2x] =0 [3sin2x - cos2x][3cos2x - sin2(x) + 1] = 0 3sin2x - cos2x = 0 or 3cos2x - sin2(x) + 1 = 0 3sin2x = cos2x or 3cos2x + cos2x = 0 3tan2x = 1 or 4cos2x = 0 tan2x = 1/3 or cos2x = 0 tan(x) = 1/√3 or tan(x) = -1/√3 or cos(x) = 0 x = 30° or 180°+30° or 180°-30° or 360°-30° or 90° or 270° x = 30° or 150° or 210° or 330° or 90° or 270° 〔共六個根。〕 2013-09-17 21:58:26 補充： 字數超額，只可以在此補充： 0° ≤ x ≤ 360°－＞0° ≤ 5x/2 ≤ 900° [留意一個cycle之中 tan y = 0 只有 y = 0°, 180°, 360°] 所以 tan(5x/2) = 0代表 5x/2 = 0° or 180° or 360° (第一個cycle) or 360°+180° or 360°+360° (第二個cycle) or 360°+360°+180° (第三個cycle) 即 5x/2 = 0° or 180° or 360° or 540° or 720° or 900°

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