The general solution of sin x - 3 sin2x + sin3x = cos x - 3 cos2x + cos3x is
视频信息
答案文本
视频字幕
Let's solve this trigonometric equation step by step. We have sin x minus 3 sin 2x plus sin 3x equals cos x minus 3 cos 2x plus cos 3x. First, we'll rearrange the equation by grouping similar terms together.
Now we apply the sum-to-product identities. For sine A plus sine B, we get 2 sine of A plus B over 2 times cosine of A minus B over 2. Similarly for cosine. Applying these to our terms: sine x plus sine 3x becomes 2 sine 2x cosine x, and cosine x plus cosine 3x becomes 2 cosine 2x cosine x.
Now we substitute our sum-to-product results back into the equation. We get 2 sine 2x cosine x minus 3 sine 2x equals 2 cosine 2x cosine x minus 3 cosine 2x. Rearranging and factoring by grouping, we can factor out the common term sine 2x minus cosine 2x, giving us the product of two factors equal to zero.
Now we solve each factor separately. For Case 1, sine 2x minus cosine 2x equals zero, which means sine 2x equals cosine 2x. This gives us tangent 2x equals 1, so 2x equals n pi plus pi over 4. For Case 2, 2 cosine x minus 3 equals zero, which would mean cosine x equals 3 over 2. But this is impossible since cosine values must be between negative 1 and 1.
Finally, we solve for x from our valid case. We have 2x equals n pi plus pi over 4. Dividing both sides by 2, we get x equals n pi over 2 plus pi over 8, where n is any integer. This is our complete general solution to the original trigonometric equation.