sin(2kπ+α)=sinα (k∈Z)

cos(2kπ+α)=cosα (k∈Z)

tan(2kπ+α)=tanα (k∈Z)

cot(2kπ+α)=cotα (k∈Z)

sec(2kπ+α)=secα (k∈Z)

csc(2kπ+α)=cscα (k∈Z)

Representation of angle in angle system;

sin (α+k 360 )=sinα(k∈Z)

cos(α+k 360 )=cosα(k∈Z)

tan (α+k 360 )=tanα(k∈Z)

cot(α+k 360 )=cotα (k∈Z)

sec(α+k 360 )=secα (k∈Z)

csc(α+k 360 )=cscα (k∈Z)

Representation of angles in arc system;

sin(π+α)=-sinα (k∈Z)

cos(π+α)=-cosα(k∈Z)

tan(π+α)=tanα(k∈Z)

cot(π+α)=cotα(k∈Z)

sec(π+α)=-secα(k∈Z)

csc(π+α)=-cscα(k∈Z)

Representation of angle in angle system;

sin( 180 +α)=-sinα(k∈Z)

cos( 180 +α)=-cosα(k∈Z)

tan( 180 +α)=tanα(k∈Z)

cot( 180 +α)=cotα(k∈Z)

sec( 180 +α)=-secα(k∈Z)

csc( 180 +α)=-cscα(k∈Z)

sin(-α)=-sinα(k∈Z)

cos(-α)=cosα(k∈Z)

tan(-α)=-tanα(k∈Z)

cot(-α)=-cotα(k∈Z)

sec(-α)=secα(k∈Z)

csc-α)=-cscα(k∈Z)

Representation of angles in arc system;

sin(π-α)=sinα(k∈Z)

cos(π-α)=-cosα(k∈Z)

tan(π-α)=-tanα(k∈Z)

cot(π-α)=-cotα(k∈Z)

sec(π-α)=-secα(k∈Z)

cot(π-α)= csα(k∈Z)

Representation of angle in angle system;

sin( 180 -α)=sinα(k∈Z)

cos( 180 -α)=-cosα(k∈Z)

tan( 180 -α)=-tanα(k∈Z)

cot( 180 -α)=-cotα(k∈Z)

sec( 180 -α)=-secα(k∈Z)

Representation of angles in arc system;

sin(2π-α)=-sinα(k∈Z)

cos(2π-α)=cosα(k∈Z)

tan(2π-α)=-tanα(k∈Z)

cot(2π-α)=-cotα(k∈Z)

sec(2π-α)=secα(k∈Z)

csc(2π-α)=-cscα(k∈Z)

Representation of angle in angle system;

sin(360 -α)=-sinα(k∈Z)

cos(360 -α)=cosα(k∈Z)

tan(360 -α)=-tanα(k∈Z)

cot(360 -α)=-cotα(k∈Z)

Seconds (360-α)= Seconds α(k∈Z)

csc(360 -α)=-cscα(k∈Z)

Representation of angles in arc system;

sin(π/2+α)=cosα(k∈Z)

cos(π/2+α)=—sinα(k∈Z)

tan(π/2+α)=-cotα(k∈Z)

cot(π/2+α)=-tanα(k∈Z)

sec(π/2+α)=-cscα(k∈Z)

csc(π/2+α)=secα(k∈Z)

Representation of angle in angle system;

sin(90 +α)=cosα(k∈Z)

cos(90 +α)=-sinα(k∈Z)

tan(90 +α)=-cotα(k∈Z)

cot(90 +α)=-tanα(k∈Z)

sec(90 +α)=-cscα(k∈Z)

csc(90 +α)=secα(k∈Z)

Representation of angles in arc system;

sin(π/2-α)=cosα(k∈Z)

cos(π/2-α)=sinα(k∈Z)

tan(π/2-α)=cotα(k∈Z)

cot(π/2-α)=tanα(k∈Z)

sec(π/2-α)=cscα(k∈Z)

csc(π/2-α)=secα(k∈Z)

Representation of angle in angle system;

sin (90 -α)=cosα(k∈Z)

cos (90 -α)=sinα(k∈Z)

tan (90 -α)=cotα(k∈Z)

cot (90 -α)=tanα(k∈Z)

sec (90 -α)=cscα(k∈Z)

csc (90 -α)=secα(k∈Z)

Representation of angles in arc system;

sin(3π/2+α)=-cosα(k∈Z)

cos(3π/2+α)=sinα(k∈Z)

tan(3π/2+α)=-cotα(k∈Z)

cot(3π/2+α)=-tanα(k∈Z)

sec(3π/2+α)=cscα(k∈Z)

csc(3π/2+α)=-secα(k∈Z)

Representation of angle in angle system;

sin(270 +α)=-cosα(k∈Z)

cos(270 +α)=sinα(k∈Z)

tan(270 +α)=-cotα(k∈Z)

cot(270 +α)=-tanα(k∈Z)

sec(270 +α)=cscα(k∈Z)

csc(270 +α)=-secα(k∈Z)

Representation of angles in arc system;

sin(3π/2-α)=-cosα(k∈Z)

cos(3π/2-α)=-sinα(k∈Z)

tan(3π/2-α)=cotα(k∈Z)

cot(3π/2-α)=tanα(k∈Z)

sec(3π/2-α)=-secα(k∈Z)

csc(3π/2-α)=-secα(k∈Z)

Representation of angle in angle system;

sin(270 -α)=-cosα(k∈Z)

cos(270 -α)=-sinα(k∈Z)

tan(270 -α)=cotα(k∈Z)

cot(270 -α)=tanα(k∈Z)

sec(270 -α)=-cscα(k∈Z)

csc(270 -α)=-secα(k∈Z)

Triple angle formula

four times the angle formula

Five-fold angle formula

Hexagonal formula

Seven-angle formula

Octagonal formula

sin8a=-8*(cosa*sina*(2*sina^2- 1)*(-8*sina^2+8*sina^4+ 1))

cos8a= 1+( 160*cosa^4-256*cosa^6+ 128*cosa^8-32*cosa^2)

tan8a=-8*tana*(- 1+7*tana^2-7*tana^4+tana^6)/( 1-28*tana^2+70*tana^4-28*tana^6+tana^8)

Nine-angle formula

sin9a=(sina*(-3+4*sina^2)*(64*sina^6-96*sina^4+36*sina^2-3))

cos9a=(cosa*(-3+4*cosa^2)*(64*cosa^6-96*cosa^4+36*cosa^2-3))

tan9a=tana*(9-84*tana^2+ 126*tana^4-36*tana^6+tana^8)/( 1-36*tana^2+ 126*tana^4-84*tana^6+9*tana^8)

Ten-fold angle formula

sin 10a=2*(cosa*sina*(4*sina^2+2*sina- 1)*(4*sina^2-2*sina- 1)*(-20*sina^2+5+ 16*sina^4))

cos 10a=((- 1+2*cosa^2)*(256*cosa^8-5 12*cosa^6+304*cosa^4-48*cosa^2+ 1))

tan 10a=-2*tana*(5-60*tana^2+ 126*tana^4-60*tana^6+5*tana^8)/(- 1+45*tana^2-2 10*tana^4+2 10*tana^6-45*tana^8+tana^ 10)

(Note: Angle A is the included angle between plane B and plane C)

(Note: Angle B is the included angle between side A and side C)

(Note: Angle C is the angle between side A and side B) Given the three sides A, B, C and the half circumference P of a triangle, then S= √[p(p-a)(p-b)(p-c)].

(p= (a+b+c)/2)

And: (a+b+c)*(a+b-c)* 1/4.

Given the angle c between two sides a and b of a triangle, S=absinC/2.

Let the three sides of a triangle be A, B and C respectively, and the radius of the inscribed circle be R.

Then the triangle area =(a+b+c)r/2.

Let the three sides of a triangle be A, B and C respectively, and the radius of the circumscribed circle be R.

Triangle area =abc/4r.

Given that the three sides of a triangle are A, B and C, then S = √ {1/4 [C2A2-((C2+A2-B2)/2) 2]} (Quadrature of Diagonal Diagrams) Note: Qin formula is equivalent to Helen formula.

| a b 1 |

S△= 1/2 * | c d 1 |

| e f 1 |

| a b 1|

| c d 1| is a third-order determinant, and this triangle ABC is in the plane rectangular coordinate system A(a, b), B(c, d), C(e, f), where | e f 1 |.

It's best to select ABC constituencies in counterclockwise order from the upper right corner, because the results obtained in this way are generally positive. If you don't follow this rule, you may get a negative value, but it doesn't matter, just take the absolute value and it won't affect the size of the triangle area!

Area formula of central line of Qin triangle

s =√[(Ma+m b+Mc)*(m b+Mc-Ma)*(Mc+Ma-Mb)*(Ma+m b-Mc)]/3

Where ma, MB and MC are the lengths of the midline of the triangle. Arcsine (-x)=- Arcsine

arccos(-x)=π-arccosx

Arctangent (-x)=- arctangent

arccot(-x)=π-arccotx

Arc sin x+ arc cos x=π/2

Arc tan x+ arc cot x=π/2