formula

Refers to a set of identities sin α+sinβ=2sin in the trigonometric function part of high school mathematics, so it should be multiplied by 2. It can also be remembered by its proof, because the two uncompensated terms are the same after expanding the two-angle sum and difference formula, resulting in a coefficient of 2, such as: cos (α-β)-cos (α+β) = [(cos α cos β+sin α sin β)-(cos α cos β-sin α sin β)] = 2 sin α sin β, so it needs to be multiplied by 2 finally.

Only trigonometric functions with the same name can add and subtract products.

No matter sine function or cosine function, only the sum and difference of trigonometric functions with the same name can be converted into products. This is mainly based on proof memory, because if it is not a trigonometric function with the same name, the product term of the sum and difference formula of two angles will have different forms, so there will be no cancellation and the same term, and it is impossible to simplify it.

The angle in the product term is divided by 2.

In the proof of the sum-difference product formula, α and β must be expressed as the sum-difference of two angles before they can be expanded. As we all know, to make the sum and difference of two angles equal to α and β respectively, these two angles should be (α+β)/2 and (α-β)/2, that is, the form of angles in the product term. Note that there is a "divide by 2" in the sum-difference product and the sum-difference formula, but the position is different; Only the sum-difference product formula has "multiply 2".

Which is the product of two trigonometric functions?

A better way to remember this is to divide it into two parts, one is whether it is a product of the same name, and the other is the trigonometric function name of "half-difference angle" (α-β)/2. Whether a product has the same name or not depends on proof and memory. Note that in the sum-difference formula of two angles, the expansion of cosine contains the product of two pairs of trigonometric functions with the same name, while the expansion of sine is the product of two pairs of trigonometric functions with different names. So the sum and difference of cosine become the product of trigonometric functions of the same name; The sum and difference of sine become the product of trigonometric functions with different names. The laws of trigonometric function names of (α-β)/2 are as follows: When sum is converted into product, it appears in the form of cos(α-β)/2; On the other hand, it is sin(α-β)/2. It is most convenient to remember this with the parity of functions. If the sum is to be multiplied, the inversion of α and β has no effect on the result, that is, if (α-β)/2 is replaced by (β-α)/2, the result should be the same, so the form of (α-β)/2 is COS (α-β)/2; The other situation can be explained similarly.

Inverse/Negative Sign in Cosine-Cosine Difference Formula

This is a special case and can be memorized by rote. Of course, there are other ways to help judge this situation, such as monotonicity of cosine function in (0, π). Because the cosine function is monotonically decreasing in this interval, when α is greater than β, cosα is less than cosβ. But at this time, the corresponding (α+β)/2 and (α-β)/2 are in the range of (0, π), and the product of their sinusoids should be greater than 0, so either put cosβ in front of cosα in turn or add a negative sign in front of the formula.