Judgment of the parity of the minimum period of trigonometric functions

Solution-shape combination method.

sinx) 24 is to flip the image of sinx under the x-axis and take it to the 24th power.

So his period is sinx, i.e., the period is

Look at parity after that, because the image in sinx (- 2,0) is below the x-axis.

After the 24th power, it turned up.

So it becomes an even function.

In summary, (sinx) 24 period is , even function.

sin4x) 2, parity analysis is the same as above, even function.

t=2π/w=2π/4=π/2

In summary(sin4x) 2 period is 2, even function.

Functions like f(x)=f(-x) are even functions, and functions like -f(x)=f(-x) are basis functions, graphically speaking, the odd function is symmetrical with respect to the center of the coordinate origin, and the even function is symmetric with respect to the y-axis, therefore, by the trigonometric function characteristics, y=sinx is the basis function, then y=sin 24x is also an odd function, and the minimum period t=2 w=2 24= 12 can be obtained by definition.

The form of the positive sign before the first transformation into x, that is, through the identity deformation into a form such as acos(bx+c), only when a trigonometric function can be transformed into acos(bx) - sine and tangent are also OK, that is, there is no c, there is parity, sine and tangent are odd functions, and cosine is an even function.

Minimum positive period = period b in the standard form of the function, e.g. 2 b for cosine, 2 b for sine, and b for tangent

Typing is not easy, such as satisfaction, hope.

The properties of the sinusoidal function are:

1. Monotonic intervals.

The sinusoidal function increases monotonically on [- 2+2k, 2+2k] and decreases monotonically on [2+2k, 3 2+2k].

2. Parity.

The sine function is an odd function.

3. Symmetry: The sinusoidal function refers to the axial symmetry of the only code x= 2+2k.

Yu does about (k,0) central symmetry.

4. Periodicity: The period of the sinusoidal function is 2.

Sinusoidal function relation:

The relationship between the product: sin = tan cos (i.e. sin cos = tan ).

cos = cot sin (i.e. cos sin = cot).

tan = sin sec (i.e. tan sin = sec).

Reciprocal relation: tan cot = 1

sinα ×cscα =1

cosα ×secα =1

f(x) is an even function.

So f(7)=f(-7) The algorithm for the minimum positive period: 1. Definition method.

The period is found directly from the definition of the periodic function.

Second, the formula method.

The following formula is used to solve the minimum positive return date of the Sankeino hunger angle function.

3. Transformation method.

For more complex trigonometric functions, the identity deformation can be converted into equal type, and then solved by the formula method.

Fourth, the least common multiple method.

The trigonometric equation composed of the algebra sum of trigonometric functions can be obtained by first finding the minimum positive period of each addition function, and then finding the least common multiple of all periods.

To judge the parity and periodicity of a function, you need to understand the basic concepts and properties of the function. Source hail blue.

The parity of a function can be judged by the following steps:

Determine the domain of the function to see if it is symmetrical with respect to the origin.

Calculate the value of the function at x=0 to see if it is 0.

If the value of the function at x=0 is 0, the parity of the function is judged based on the relationship between f(x) and f(-x). If f(x)=f(-x), then the function is even; If f(x)=-f(-x), then the function is odd.

The periodicity of a function can be judged by the following steps:

Look at the expression of the function to see if it contains a form similar to sin(nx) or cos(nx).

If there is a form similar to sin(nx) or cos(nx), then according to the definition of a periodic function, the function is a periodic function.

Determine the period, i.e., what the period of the function is.

In summary, judging the parity and periodicity of a function requires mastering the basic concepts and properties of the function, and making judgments through the analysis of the function expression and definition domain.

Trigonometric function at the top of the small positive period formula:

1. The minimum positive period t=2|of y=asin(x+)h or y=acos(x+)h

2. The minimum positive period t= of y=atan(x+)h or y=acot(x+)h

3、y=|sinωx|or y=|cosωx|The minimum positive period t= <>

4、y=|tanωx|or y=|cotωx|There are five ways to find the minimum positive period of a trigonometric function t= the definition method, the public chain spring method, the transformation wisdom jujube method, the least common multiple method, and the image method.

The parity of trigonometric functions is:1. y=sinx1. Parity: odd functions2. Image nature:

Central symmetry: Symmetry with respect to the point (k,0).Axisymmetry: Symmetry with respect to x=k2.

2. y=cosx

1. Parity: even function.

2. Image nature:

Central symmetry: Symmetry with respect to the point (k 2,0).

Axisymmetry: Symmetry with respect to x=k.

3. y=tanx

1. Parity: odd functions

2. Image nature:

Central symmetry: Symmetry with respect to the point (k 2,0).

The function operation algorithm is used to determine the parity of functions.

Odd Functions Odd Functions Odd Functions.

Even Function, Even Function, Modular, Even, Even, Function, Figure, Even, Function, Figure, Figure, Figure

Odd Functions Odd Functions Even Functions.

Even Functions Even Functions Even Functions.

Even Functions, Odd Functions, Odd Functions, Code Drawbacks.

(1) The monotonicity of odd functions in the symmetry interval is the same, and the monotonicity of even functions in the symmetry interval is opposite;

2) Parity is a special symmetry, that is, parity pushes out symmetry, and symmetry does not push out parity. Periodicity and parity, periodicity and symmetry cannot be pushed out of each other.

3) Periodic functions may or may not be monotonionic in a cycle, and monotonic functions generally do not have periodicity. That is, periodicity and monotonicity cannot be pushed out of each other.

First of all, for y=asin(wx+q)+b, we must first know that a is a positive number (if it is a negative number, the image is exactly upside down, and it is found to be exactly the opposite), and then according to the property of the sinusoidal function, in [- 2

2k, 2+2k] is increased, and - 2+2k +2k can be found, and the periodicity is found with the formula t=2 w, note that it is not necessarily 2 as the dividend, it should be determined according to the period of the original standard function, and the tangent function is

Judged by definition, it is best.

f(-x) = f(x) is an even function.

f(-x)=-f(x) is an odd function.

f(x+t)=f(x) is a periodic function, and the period is t

Function parity.

If f(-x)=-f(x) is an odd function, f(-x)=f(x) is an even function. The periodic function is generally observed first, and different types of periodic algorithms are different.

In general, for the function f(x).

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

2) If there is f(-x)=f(x) for any x in the function definition field, then the function f(x) is called an even function.

3) If for.

The function is defined within the domain.

f(-x)=-f(x) and f(-x)=f(x), (x d, and d is symmetrical with respect to the origin. Then the function f(x) is both odd and even, and is called both odd and even.

4) If for.

The function defines the domain.

f(-x)=-f(x) and f(-x)=f(x) cannot be true, then the function f(x) is neither odd nor even, and is called a non-odd and non-even function.

Note: Odd and evenness are integral properties of a function, for the entire defined domain.

The domain of an odd and even function must be symmetrical with respect to the origin, and if the domain of a function is not symmetrical with respect to the origin, then the function must not be parity.

Let f(x)=|sinx|+|cosx|, so, f(-x)=|sin-x|+|cos-x|=|sinx|+|cosx|=f(x), as you can see, it is an even function! f(x+π)=|sin(x+π)cos(x+π)=|sinx|+|cosx|, the minimum positive period is