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• 1 、Find the range of y=sina+cosa+2sina*cosa

Find the range of y=sina+cosa+2sina*cosa

Get: sin (a+a)=sinacosa+cosasina=2sinacosa. Namely: sin2a=2sinacosa. Universal formula derivation: Let tan (A/2)=t. sinA=2t/（1+t^2） （A≠2kπ+π ，k∈Z）。tanA=2t/（1-t^2） （A≠2kπ+π
，k∈Z） 。cosA=（1-t^2）/（1+t^2） （A≠2kπ+π k∈Z）
。

- (ap+cp)=mk-mp. Since me=mk, the above value =me-mp= ep0, that is, the sum of the vertical and horizontal coordinates of point k is greater than the sum of the vertical and horizontal coordinates of other points on the arc, and sina+cosa has a maximum value at point k. It is easy to know that bk=dk=2/2 square root 2, so the maximum value is square root 2. Similarly, in the third quadrant, sina+cosa has a minimum minus square root of 2. At this time, we can get that the range is (negative square root sign 2, square root sign 2), and the answer is over.

When a= 322 °, y=sinacosa^2=0.3848985, when a= 321 °, y=sinacosa^2=0.3848999, When a= 321 °, max y=sinacosa^2=0.3848999.

First, from the square formula, we know that (sina + cosa)^2 = sina^2 + 2 * sina * cosa + cosa ^2. Given the value of sina + cosa, the value of sina^2 + cosa^2 + 2 * sina * cosa can be obtained by squaring.

When we consider the range of values for sina+cosa, we can rewrite it as a single-variable trigonometric function. Specifically, we have sina+cosa=√2sin (a+π/4). The key here is to realize that the value range of sin (a+π/4) is [-1, 1], because this is the basic property of all sinusoidal functions. Therefore, by multiplying the sine function by √2, we get a range of values for sina+cosa as [-√2, √2].