Let tan(a/2)=t

sina=2t/( 1+t^2)

cosa=( 1-t^2)/( 1+t^2)

tana=2t/( 1-t^2)

2. Auxiliary angle formula

asint+bcost=(a^2+b^2)^( 1/2)sin(t+r)

cosr=a/[(a^2+b^2)^( 1/2)]

sinr=b/[(a^2+b^2)^( 1/2)]

tanr=b/a

3. Triple angle formula

sin(3a)=3sina-4(sina)^3

cos(3a)=4(cosa)^3-3cosa

tan(3a)=[3tana-(tana)^3]/[ 1-3(tana^2)]

4. Sum and difference of products

Sina * cosb =[sin(a+b)+sin(a-b)]/2

cosa * sinb =[sin(a+b)-sin(a-b)]/2

cosa * cosb =[cos(a+b)+cos(a-b)]/2

Sina * sinb =-[cos(a+b)-cos(a-b)]/2

5. Sum and difference of products

Sina+sinb = 2 sin[(a+b)/2]cos[(a-b)/2]

Sina-sinb = 2sin[(a-b)/2]cos[(a+b)/2]

cosa+cosb = 2cos[(a+b)/2]cos[(a-b)/2]

Cosa-cosb =-2 sin [(a+b)/2] sin [(a-b)/2] vector formula:

1. unit vector: unit vector a0= vector a/| vector a|

2.P(x, y) then vector OP=x vector i+y vector j.

| vector OP|= root sign (x square +y square)

3.P2(x2，y2)

Then the vector p 1p2 = {x2-x 1, y2-y 1}

| vector P 1P2|= radical sign [(x2-x 1) square +(y2-y 1) square]

4. Vector A = {x 1, x2} Vector B = {x2, y2}

Vector a* vector b=| vector a | | vector b | * cos α = x1x2+y1y2.

Cosα= vector a* vector b/| vector a|*| vector b|

(x 1x2+y 1y2)

= ————————————————————

Root number (x 1 square +y 1 square) * root number (x2 square +y2 square)

5. Space vector: same as above.

(Hint: Vector A = {x, y, z})

6. Necessary and sufficient conditions:

If vector a⊥ vector b

Then vector a* vector b=0

If vector a// vector b

Then vector a* vector b =+| vector a|*| vector b= |

Or x 1/x2=y 1/y2.

7. Vector A Vector b| Square

= | Vector a| Square+| Vector b| Square 2 Vector a* Vector B.

= (Vector A, Vector B) squared

a & gtb，b & gtc = & gta & gtc； a & gtb = & gta+c & gt； b+ c； A>b, c>0 = & gtac & gt BC; A>b, c<0 = & gtac & lt BC; a & gtb & gt0，c & gtd & gt0 = & gtac & gtBD； a & gtb，ab & gt0 = & gt 1/a & lt； 1/b； a & gtb & gt0 = & gta^n>； b^n； Basic inequality: (root number ab)≤(a+b)/2, then it can be changed to A 2-2ab+B 2 ≥ 0A 2+B 2 ≥ 2ab. There are two! One is ||| a |-| b | ≤| a-b |≤| a |+b | B | The other is |||| A |-B |≤| A+B |≤| Prove the available vector, take A and B as vectors, and use the difference between the two sides of the triangle to be smaller than the first one. 1. Definition of parabola: The trajectory of a point with the same distance from a point (F) on the plane and a fixed line (L) is called parabola. This fixed point F is called the focus of parabola, and this fixed line L is called the directrix of parabola. It should be emphasized that the point F is not on the straight line L, otherwise the trajectory is a straight line passing through the point F and perpendicular to L, not a parabola. 2. Parabolic equation For the above four equations, we should pay attention to their laws: which axis is the symmetry axis of the curve, and the terms in the equation are linear terms; There is a plus sign in front of the first term, and the opening direction of the curve is in the positive direction of X axis or Y axis; When the first term is preceded by a negative sign, the opening direction of the curve is the negative direction of the X axis or the Y axis. 3. The geometric properties of parabola take the standard equation y2=2px as an example (1): x ≥ 0; (2) Symmetry axis: the symmetry axis is y=0, which can be seen from the equation and the image; (3) Vertex: O (0 0,0), Note: Parabola is also called centerless conic curve (because there is no center); (4) Eccentricity: e= 1, because E is a constant, the shape change of parabola is determined by P in the equation; (6) formula of focal radius: a point on the parabola is P(x 1, y 1), and f is the focus of the parabola. For four kinds of parabolas, the focal radius formulas are (p > 0) respectively. (7) Focal chord length formula: For the chord length passing through the parabolic focal point, the chord length formula can be derived by using the focal radius formula. Let the chord of the parabola y2 = 2px (p > o) passing through the focus f be AB, A(x 1, y 1), B(x2, y2), and the inclination of AB be α, then ①|AB|=x 1+x2+p, and the above two formulas are only (8) Relationship between straight line and parabola: After the combination of straight line and parabola equation, the unary quadratic equation is obtained: ax2+bx+c=0. When a≠0, the positional relationship between them is the same as that of ellipse and hyperbola, so the discrimination method can be used; However, if a=0, the straight line is the symmetry axis of parabola or a straight line parallel to the symmetry axis. At this time, the straight line intersects with the parabola, but there is only one thing in common. (9) The tangent y2 of the parabola = 2px: ① If the point P(x0, y0) is on the parabola, then y0y = p (x+x0); (10) parametric equation Understand the concept of parametric equation, understand the geometric meaning or physical meaning of parameters in some commonly used parametric equations, and master the mutual conversion method between parametric equation and ordinary equation. According to the given parameters, the parameter equation is established according to the conditions. 1.y = c (c is a constant) y' = 0 2. y = x ny' = NX(n- 1) 3 . y = a x y ' = a XL nay = e x y ' = e . x 5 . y = sinx y ' = cosx 6 . y = cosx y ' =-sinx 7 . y = tanx y'= 1/cos^2x 8 . y = cotx y'=- 1/sin^2x 9 . y = arcsinx y ' = 1/√65438+