1. Trigonometric and inverse trigonometric formulas
1. Basic relations of trigonometric functions
(1) The product on the diagonal is 1: cscx = 1 / sinx , secx = 1 / cosx , cotx = 1 / tanx
(2) A vertex is equal to the product of two adjacent vertices: tanx = sinx / cosx , cotx = cosx / sinx
(3) The sum of the squares of the two vertices on the shaded triangle is equal to the square of the lower vertex:
sin²x + cos²x =1 , 1 + tan²x = sec²x , 1 + cot²x = csc²x
2. Induction formula
For the trigonometric function value of (kπ / 2) ± x (k ∈ Z )
(1) See the quadrants for symbols: Treat x as an acute angle of 0, and see (kπ / 2) ± In the second quadrant, only sin is positive, in the third quadrant, tan cot is positive, and in the fourth quadrant, only cos is positive)
(2) Invariance from odd to even: when k is an even number, the function name does not change; when k is an odd number, the function name changes to the corresponding co-function value, that is, sin->cos, cos->sin, tan-> cot, cot->tan
[example]
sin[(π / 2) - x] = cosx
cos[(π / 2) - x] = sinx
sin[(π / 2) + x] = cosx
cos[(π / 2) + x] = -sinx
sin(π - x) = sinx
cos(π - x) = -cosx
sin(π + x) = -sinx
cos(π + x) = -cosx
3. Double angle formula
sin2x = 2sinxcosx , cos2x = cos²x - sin²x = 2cos²x - 1 = 1- 2sin²x
tan2x = 2tanx / (1 - tan²x) , cot2x = (cot²x - 1) / 2cotx
4. Half-width formula (falling formula)
sin²x = 1/2(1 - cos2x) , cos²x = 1/2(1+cos2x)
5. Sum and difference formula
sin(x ± y) = sinxcosy ± cosxsiny ,
cos(x ± y) = cosxcosy ∓ sinxsiny ,
tan(x ± y) = (tanx ± tany) / (1 ∓ tanxtany) ,
cot(x ± y) = (cotxcoty ∓ 1) / (cotx ± coty) ,
asinx + bcosx = √ (a²+b²) sin [x+arctan (b/a)] (auxiliary angle formula)
6. Integration and difference formula
Formula: positive remainder half positive positive remainder half remainder positive and negative half remainder minus remainder
sinxcosy = 1/2[sin(x+y) + sin(xy)]
cosxcosy = 1/2[cos(x+y) + cos(x-y)]
sinxsiny = -1/2[cos(x+y) - cos(xy)]
7. Sum-difference product formula
Formula: positive + positive two plus plus plus plus plus minus two plus plus plus plus minus two plus plus plus plus minus two plus plus plus minus two plus minus plus minus two plus plus
sinx+siny = 2sin (x+y)/2 cos (xy)/2
sinx-siny = 2cos (x+y)/2 sin (xy)/2
cosx+cosy = 2cos (x+y)/2 cos (x-y)/2
cosx-cosy = -2sin (x+y)/2 sin (xy)/2
8. Universal formula
若t = tan(x/2) (-π<x<π)
Then sinx = 2t/(1+t²) , cosx = (1-t²)/(1+t²)
9. Basic relations of inverse trigonometric functions
arcsinx + arccosx = π/2 (-1<=x<=1) , arctanx + arccotx = π/2 (-∞<x<+∞)
2. Algebra and Equations
1. Exponential logarithmic algorithm
aᵐ·aⁿ = aᵐ⁺ⁿ , (aᵐ)ⁿ = a(ᵐⁿ) , (ab)ⁿ = aⁿbⁿ , a^-1 = 1/a , ⁿ√a = a^(1/n) , a⁰=1 , √a² = |a| ,
lna + lnb =ln(ab) , ln(a/b) = lna - lnb , lnaⁿ =nlna , a = e^lna
2. Commonly used sequence
(1) Arithmetic sequence: the first term is a1, and the common difference is d (d!=0)
General term: an=a1+ (n-1)d
Sum of the first n terms: Sn=n(a1+an)/2
(2) Geometric sequence: the first term is a1, and the common ratio is q (q!=0)
General term: an=a1q^(n-1)
Sum of first n terms:
Sn= na1 q=1 ,
= [a1(1-q^n)] / (1-q) q!=1
3. Factoring formula
(a ± b)² = a² ± 2ab + b²
(a ± b)³ = a³ ± 3a²b + 3ab² ± b³
a² - b² = (a + b)(a - b)
a³ - b³ = (a - b)(a² + ab + b²)
a³ + b³ = (a + b)(a² - ab + b²)
3. Commonly used inequalities
x-1 < [x] <= x < [x]+1
||a|-|b|| <= |a±b| <= |a|+|b|
x/(1+x) < ln(1+x) <x (x>0)
e^x >=x+1
sinx < x (x>0)
sinx < x < tanx (0<x<π / 2)
2/[(1/a)+(1/b)] <= √(ab) <= (a+b)/2 <= √[(a²+b²)/2] (a,b>0)
4. Factorial and double factorial
1. Factorial: represents the product of a positive integer and all positive integers smaller than it.
example:
Rule 0! =1
3！=3*2*1
5！=5*4*3*2*1
2. Double factorial: The double factorial of a positive integer represents the product of all positive integers that do not exceed this positive integer and have the same parity as it.
example:
1！！=1，
2！！=2，
4！！=2*4=8，
5！！=1*3*5=15，
6！！=2*4*6=48