Unified deformation formula How to calculate asin α + bsin α = (radical a ^ 2 + B ^ 2) sin (x + α) x

Unified deformation formula How to calculate asin α + bsin α = (radical a ^ 2 + B ^ 2) sin (x + α) x

tanx=b/a

φ in the formula of mathematical unity
The unification formula is that if y = asinx plus bsinx plus constant, then f (x) = under the root sign (the square of a plus the square of B) multiplied by sin (x plus φ) plus constant. How can I get the value of φ? I know that Tan φ equals B / A, and I can get it by computer, but is there any other way? For example, B / a equals minus 2 / 7, and I can get the value of 15. Several, infinite decimal by computer. What can I do?

It is well known that the trigonometric function in the form of asin α + boos α can be transformed into the form of √ [a ^ 2 + B ^ 2] sin (α + φ). Here, the function name must be sine and cosine, and the angle must be the same. φ is called auxiliary angle. The quadrant of φ is determined by the sign of a and B, and the value of φ angle is usually determined by Tan φ = B / A
If we know that Tan is equal to B / A, how can we find it?
1. Using computer, we can find out,
2. Other methods:
First, the quadrant of φ is determined by the symbols of a and B;
Then, from the value of | B / a | to the tangent value table in the four digit mathematical table, the acute angle corresponding to φ is obtained,
Finally, from the quadrant where φ is, the induced formula can be used to get φ
For example:
B / a equals minus 2 / 7
It is found that the quadrant of φ is two quadrants
It can be expressed as π - arctan2 / 7
From | - 2 / 7 | = 2 / 7 ≈ 0.2857
According to the tangent value table in the four digit mathematical table, the acute angle corresponding to φ is 16 ° 36 ′,
By using the induction formula, it can be obtained that φ = 180 ° - 16 ° 36 ′ = 163 ° 24 ′
Infinite decimal, how to do?
It can be solved by approximate value

Mathematical plane geometry formula

1. There is only one straight line passing through two points. 2. The shortest line segment between two points. 3. The complementary angles of the same angle or equal angle are equal. 4. The complementary angles of the same angle or equal angle are equal. 5. There is only one straight line passing through one point and the known straight line is vertical. 6. Among all the line segments connected by a point outside the straight line and each point on the line, the shortest vertical line segment is 7. The axiom of parallel passing through the straight line