Arbitrary angle (1) In geometry, an angle is a geometric object composed of two rays with a common endpoint. These two rays are called the edges of an angle, and their common endpoint is called the vertex of the angle.
(2) In the two-dimensional Cartesian coordinate system, the angle is generally based on the positive direction of the X axis. If it rotates to the positive direction of the Y axis, its angle is positive, and if it rotates to the negative direction of the Y axis, its angle is negative. If the two-dimensional Cartesian coordinate system is also the X-axis to the right and the Y-axis to the right, then the counterclockwise rotation corresponds to a positive angle and the clockwise rotation corresponds to a negative angle.
(3) trigonometric function value of special angle
Commonly used formulas of trigonometric functions Half-angle formulas of trigonometric functions
sin(A/2)= √(( 1-cosA)/2)
cos(A/2)= √(( 1+cosA)/2)
tan(A/2)=√(( 1-cosA)/(( 1+cosA))
Double angle formula of trigonometric function
Sin2A=2SinA*CosA
cos2a=cosa^2-sina^2= 1-2sina^2=2cosa^2- 1
tan2A=(2tanA)/( 1-tanA^2)
Sum and difference formula of two angles of trigonometric function
sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-cossinB
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)
tan(A-B)=(tanA-tanB)/( 1+tanA tanB)
Sum and difference of trigonometric function products
Sina sinb =-[cos(A+B)-cos(A-B)]/2
cosAcosB=[cos(A+B)+cos(A-B)]/2
sinAcosB=[sin(A+B)+sin(A-B)]/2
cosAsinB=[sin(A+B)-sin(A-B)]/2
Sum-difference product of trigonometric functions
sinA+sinB = 2sin[(A+B)/2]cos[(A-B)/2]
sinA-sinB = 2cos[(A+B)/2]sin[(A-B)/2]
cosA+cosB = 2cos[(A+B)/2]cos[(A-B)/2]
cosA-cosB =-2 sin[(A+B)/2]sin[(A-B)/2]
tanA+tanB = sin(A+B)/cosa cosb = tan(A+B)( 1-tanA tanB)
tanA-tanB = sin(A-B)/cosa cosb = tan(A-B)( 1+tanA tanB)
Trigonometric function theorem (1) sine theorem
In any △ABC, the side lengths of angles A, B and C are respectively A, B and C, the radius of the circumscribed circle of the triangle is R, and the diameter is D. Then it is: a/sinA=b/sinB=c/sinC=2r=D(r is the radius of the circumscribed circle and D is the diameter).
In a triangle, the ratio of sine to diagonal of each side is equal, and the ratio is equal to the diameter (twice radius) length of the circumscribed circle of the triangle.
(2) Cosine theorem
For any triangle, the square of any side is equal to the sum of the squares of the other two sides MINUS the product of the cosine of the angle between these two sides and them.
For a triangle with sides A, B and C and corresponding angles A, B and C, there are:
①a？ =b？ +c？ -cosA in 2000 BC
②b？ =a？ +c？ -2ac cosB；
③c？ =a？ +b？ -2ab cosC .
It can also be expressed as:
①cosC=(a？ +b？ -c？ )/2ab；
②cosB=(a？ +c？ -B? )/2ac；
③cosA=(c？ +b？ -a？ )/2 BC.
(3) Tangent theorem
In a triangle, the quotient obtained by dividing the sum of any two sides by the difference between the first side and the second side is equal to the tangent of half the sum of the diagonals of these two sides divided by the tangent of the difference between the diagonals of the first side and the second side.
For a triangle with sides A, B and C and corresponding angles A, B and C, there are:
①(A-B)/(A+B)=[tan(A-B)/2]/[tan(A+B)/2]；
②(B-C)/(B+C)=[tan(B-C)/2]/[tan(B+C)/2]；
③(C-A)/(C+A)=[tan(C-A)/2]/[tan(C+A)/2]。