Sinθ=y/r

Cosine function cosθ=x/r

Tangent function tanθ=y/x

Cotangent function cotθ=x/y

Secθ secθ=r/x

Cotangent function csθ= r/y

2. Square relation of trigonometric function with the same angle:

sin^2(α)+cos^2(α)= 1

tan^2(α)+ 1=sec^2(α)

cot^2(α)+ 1=csc^2(α)

3. The relationship between the products of trigonometric functions with the same angle:

sinα=tanα*cosα

cosα=cotα*sinα

tanα=sinα*secα

cotα=cosα*cscα

secα=tanα*cscα

csα= secα* cotα

4. The reciprocal relationship between trigonometric functions with the same angle:

tanα cotα= 1

sinα cscα= 1

cosα secα= 1

5. The basic steps of finding monotonicity of function by derivative: ① finding the domain of function yf(x); ② Find the derivative of f(x); ③ Solve the inequality f(x)0, and define the uninterrupted interval of the solution set on the domain as the increasing interval; ④ If the inequality f(x)0 is solved, the uninterrupted interval of the solution set on the domain is a decreasing interval.

Conversely, we can also use derivatives to solve related problems (such as determining the range of parameters) through the monotonicity of functions: let function yf(x) be derivable in the interval (a, b),

(1) If the function yf(x) is the increasing function in the interval (a, b), then f(x)0 (where the x value of f(x)0 does not constitute the interval).

(2) If the function yf(x) is a subtraction function in the interval (a, b), then f(x)0 (where the x value of f(x)0 does not constitute the interval).

(3) If the function yf(x) is a constant function in the interval (a, b), then f(x)0 holds.

6. Find the extreme value of the function:

Let the function yf(x) be defined in x0 and its vicinity. If all points near x0 have f(x)f(x0) (or f(x)f(x0)), it is said that f(x0) is the minimum (or maximum) of the function f(x).

The extreme value of differentiable function can be obtained by studying the monotonicity of function. The basic steps are as follows:

(1) determines the domain of the function f(x).

(2) Find the derivative of f(x).

(3) Find all the real roots of the equation f(x)0, x 1x2xn, divide the domain into several cells in sequence, and list the changes of the values of f(x) and f(x) when x changes.

(4) Look up the sign of f(x) and judge the extreme value from the table.

7. Find the value and minimum value of the function:

If the function f(x) has x0 in the domain I, so that there is always f(x)f(x0) for any xI, it is said that f(x0) is the value of the function in the domain. The extreme value of the function in the definition domain is not necessarily, but the maximum value in the definition domain is.

Step of finding the value and minimum value of the function f(x) in the interval [a, b]: (1) Find the extreme value of f(x) in the interval (a, b).

(2) Compare the extreme value obtained in the first step with f(a) and f(b) to obtain the value and minimum value of f(x) in the interval [a, b].

8. Solve the related problems of inequality:

The scope of (1) inequality problem (absolute inequality problem) can be considered.

When the range of f(x)(xA) is [a, b],

The necessary and sufficient condition for the inequality f(x)0 is f(x)max0, i.e B0;

The necessary and sufficient condition for the inequality f(x)0 is f(x)min0, which is a0.

When the range of f(x)(xA) is (a, b),

The necessary and sufficient condition for inequality f(x)0 to be constant is B0; The necessary and sufficient condition for inequality f(x)0 to be constant is a0.

(2) Proving the inequality f(x)0 can be transformed into proving f(x)max0, or using the monotonicity of function f(x) to prove f(x)f(x0)0.

9. Definition of parity:

Generally, for the function f(x)

(1) If any x in the function definition domain has f (-x) =-f(x), then the function f(x) is called odd function.

(2) If any x in the function definition domain has f (-x) = f(x), the function f(x) is called an even function.

(3) If f (-x) =-f(x) and f (-x) = f (x) are true for any x in the function definition domain, then the function f(x) is both a odd function and an even function, which is called an even-even function.

10, rational number multiplication rule: (1) Multiply two numbers, the same sign is positive, the different sign is negative, and the multiplication is absolute.

(2) Any number multiplied by zero gets zero.

(3) Several factors are not zero, and the sign of the product is determined by the number of negative factors. Odd numbers are negative and even numbers are positive.