How to calculate Tan α = 2,2 / 3sin & sup2; α + 1 / 4cos & sup2; α

How to calculate Tan α = 2,2 / 3sin & sup2; α + 1 / 4cos & sup2; α

From the trigonometric function formula: Tan α = sin α / cos α: 2 / 3sin & # 178; α + 1 / 4cos & # 178; α = 2 / 3cos & # 178; α Tan & # 178; α + 1 / 4cos & # 178; α = (2 / 3tan & # 178; α + 1 / 4) cos & # 178; α = 35 / 12cos & # 178; α; from the trigonometric function formula: Sin & # 178; α + cos & # 178

It is known that two adjacent vertices a, C, B and D on the ellipse x = 4cos θ, y = 5sin θ (with θ as the parameter) are two moving points on the ellipse, and they are divided into two parts
On both sides of the line AC, find the maximum area of quadrilateral ABCD

See the picture for details

It is known that two adjacent vertices a, C, B and D on the ellipse x = 4cos, y = 5sin are two moving points on the ellipse, and they are on the straight line respectively
Given that two adjacent vertices a, C, B and D on the ellipse x = 4cos are two moving points on the ellipse, and they are on both sides of the straight line AC respectively, the maximum area of the quadrilateral ABCD is obtained

Ask for more points!
A. Let C be an ellipse, x = 4cos, y = 5sin, two adjacent vertices, a (0,5), C (4,0). The center of the ellipse is (0,0)
Please draw a picture
B. Let B be on the short arc between AC and d be on the long arc between AC
Let B be (4cosp, 5sinp), d be (4cosq, 5sinq), p be between 0 and 90 degrees, and Q be between 90 and 360 degrees
To maximize the area of the quadrilateral ABCD, point D should be in the third quadrant, so q is between 180 and 270 degrees,
The ad line intersects the x-axis at point E, and the CD line intersects the y-axis at point F,
The area of quadrilateral ABCD is the sum of four triangles, which are triangle AOB, triangle BOC, triangle AEO and triangle CED
S triangle AOB = 4cosp * 5 / 2 = 10cosp, s triangle BOC = 5sinp * 4 / 2 = 10sinp,
According to the linear formula, the coordinate of E is (4cosq / 1-sinq, 0),
S triangle AEO = 5 * | (4cosq / 1-sinq) | / 2 = - 10cosq / 1-sinq,
S triangle CED = 5 | SINQ | * (4-4cosq / (1-sinq)) / 2 = 5 | SINQ | * (2-2cosq / (1-sinq))
=-10sinQ+10sinQcosQ/(1-sinQ)
S triangle AOB + s triangle BOC = 10cosp + 10sinp = 10 radical 2Sin (* + P)

Y = 3sin (x - π / 3), y = - 3, the value set of X

x-π/3=2kπ+3π/2
x=2kπ+3π/2+π/3
x=2kπ+9π/6+2π/6
x=2kπ+11π/6