Odd and even unchanged, what does the symbol mean by looking at the quadrant

In the induction formula, you can use this formula if the angle of the difference is 90 degrees, which is an integer multiple of the disect. Typically include: sin(90°- = cos sin(90°+=cos cos(90°- = sin cos(90°+ = - sin sin(270°- = - cos sin(270°+ = - cos cos(270° + = - sin sin(180°+ = sin sin(180°+ = - sin cos(180° + = - sin cos(180° - = - cos cos(180° + = - cos cos(180° + = - - cos sin(360°- = - sin sin(360°+ = sin cos(360°- = cos cos(360°+ = cos If it is an odd multiple of 90 degrees, the function name should be changed (sin and cos, tan and cot are interchanged), and the even multiples are unchanged.

As for the symbol, consider the angle of your variable as the angle of the first quadrant, and see if it is positive or negative after the operation.

Finally, the induction formula is summarized.

Odd and even unchanged, and the sign looks at the quadrant is the mantra of the induction formula.

Odd and even are invariant (in the case of k, k is odd or even), and the sign is seen as the quadrant (looking at the original function, and can be seen as an acute angle). The symbol on the right side of the equation is the sign of the original trigonometric value of the quadrant where -180° 360°- is remembered when it is regarded as an acute angle, the angle k·360° + k z), -180° 360°- The sign of the original trigonometric value of the quadrant is remembered: the horizontal induced name does not change;symbol to see the quadrant.

How to judge the symbols of various trigonometric functions in the four quadrants, you can also remember the mantra "one is perfect;."Two sine (cosecant);Three-by-two cuts;Four cosines (secant)".

When the odd and even does not change, the case of plus and minus signs is not considered for the time being:

1. When k is an odd number, the point p on the terminal edge'( y, x) is exactly the opposite of the horizontal and longitudinal coordinates of the point p(x,y) on the original terminal edge, so the corresponding trigonometric ratio should be changed;

2. When k is even, the point p on the terminal edge'The horizontal and longitudinal coordinates of ( x, y) and the point p(x,y) on the original terminal edge do not change, so the corresponding trigonometric ratio does not change;

Symbol to see the quadrant: When using this formula, it is assumed that the original angle is an acute angle, because any trigonometric ratio of an acute angle is positive, so when judging the plus and minus signs, there is no need to consider the positive and negative situation of the trigonometric ratio itself.

For the trigonometric value of k 2 (k z), when k is an even number, the value of the function with the same name is obtained, i.e., the function name does not change; When k is an odd number, the corresponding cofunction value is obtained, i.e., sin cos; cos→sin;tan→cot,cot→tan.(odd and even unchanged) and then precede it with a sign that treats as the value of the original function when it is considered an acute angle. The trigonometric value of any angle in the first quadrant is "+", only the cosine in the second quadrant, and the cosecant is "+", and all the rest are, "only the tangent and the cotangent function in the third quadrant are" +, and the chord function is, "only the cosine and secant in the fourth quadrant, and the secant is" + "all the rest are" ".

Odd and even unchanged, sign to see quadrant" is a trigonometric function induction formula.

The phrase "odd and even unchanged" is for k, which refers to k taking odd or even numbers; "Symbolic quadrants" means judging positive and negative based on the original function, and should be regarded as acute angles. Take cos(270°- sin as an example, 270° is an odd number, so cos becomes sin; Change the grip and 270°- is the third quadrant angle, the cosine of the third quadrant angle.

is negative, so the right side of the equation is a minus.

Trigonometric inducing formula formula.

Odd and even unchanged, the symbol looks at the quadrant" can be understood as:

The trigonometric value of any angle in the first quadrant.

All of them are "+", and in the second quadrant, only the sine and cosecant are "+", and their nucleus is "-".

There is only tangent in the third quadrant.

and cotangent are "+" and the rest of the functions are "-".

In the fourth quadrant there is only secant.

And cosine is "+" and all the rest are "-".

Commonly used induction formulas.

Odd and even are invariant, and the sign looks at the quadrant is the mantra of the trigonometric function induction formula. The induction formula of trigonometric function refers to the formula that uses periodicity to convert a trigonometric function with a relatively large angle into a trigonometric function with a relatively small angle in the trigonometric function.

Equation 1: The values of the same trigonometric function for angles with the same end edge are equal

Let be any acute angle, the expression of the angle under the radian system:

sin(2kπ+αsinα(k∈z)

cos(2kπ+αcosα(k∈z)

tan(2kπ+αtanα(k∈z)

cot(2kπ+αcotα(k∈z)

Equation 2: The relationship between the trigonometric value of + and the trigonometric value of

Let be an arbitrary angle, a representation of the angle under the radian system:

sin(π+sinα

cos(π+cosα

tan(π+tanα

cot(π+cotα

Equation 3: The relationship between the trigonometric value of an arbitrary angle and -

sin(-αsinα

cos(-αcosα

tan(-αtanα

cot(-αcotα

Equation 4: Using Equation 2 and Equation 3, we can get the relationship between - and the trigonometric value of

sin(π-sinα

cos(π-cosα

tan(π-tanα

cot(π-cotα

Equation 5: Using Equation 1 and Equation 3, we can get the relationship between the trigonometric values of 2 - and

sin(2π-αsinα

cos(2π-αcosα

tan(2π-αtanα

cot(2π-αcotα

Equation 6: The relationship between the trigonometric values of 2 and

1) The relationship between the trigonometric value of 2+ and .

sin(π/2+α)cosα

cos(π/2+α)sinα

tan(π/2+α)cotα

cot(π/2+α)tanα

2) 2- The relationship between the trigonometric value and .

sin(π/2-α)cosα

cos(π/2-α)sinα

tan(π/2-α)cotα

cot(π/2-α)tanα

3) The relationship between the trigonometric values of 3 2+.

sin(3π/2+α)cosα

cos(3π/2+α)sinα

tan(3π/2+α)cotα

cot(3π/αtanα

4) 3 2- The relationship between trigonometric values.

sin(3π/2-α)cosα

cos(3π/2-α)sinα

tan(3π/2-α)cotα

cot(3π/2-α)tanα

"Odd and even are invariant, and the sign looks at the quadrant: the odd and even are unchanged (for k, it means that k is taken as an odd or even number), and the sign looks at the quadrant (looking at the original function, and at the same time can be regarded as an acute angle).

The symbol on the right side of the equation is the sign of the original trigonometric value of the quadrant where -180° 360°- is remembered when it is regarded as an acute angle, the angle k·360° + k z), -180° 360°- The sign of the original trigonometric value of the quadrant is remembered: the horizontal induced name does not change;symbol to see the quadrant.

In the induction formula trigonometric function, the trigonometric function uses periodicity to convert a trigonometric function with a relatively large angle into a trigonometric function with a relatively small angle. There were six groups of induction formulas, with a total of 54 formulas. The trigonometric value of any angle in the first quadrant is "+", and in the second quadrant, only the sine and cosecant are "+", and all the rest are " ".

In the third quadrant only tangent and cotangent are "+" and the rest of the functions are " "In the fourth quadrant only secant and cosine are "+" "All the rest are" "One perfect sine, two sine, three double tangents, four cosine,

Trigonometric functions are generally used to calculate the edges of unknown lengths and unknown angles in triangles, and have a wide range of uses in navigation, engineering, and physics.

In addition, using trigonometric functions as a template, a similar class of functions can be defined, called hyperbolic functions. The common hyperbolic function is also known as hyperbolic sine function, hyperbolic cosine function, and many more.

Trigonometric functions (also known as circular functions) are functions of angles; They are important in studying triangles and modeling periodic phenomena and many other applications.

Trigonometric function is usually defined as the ratio of the two sides of a right triangle containing this angle, and can also be defined equivalently as the length of various line segments on a unit circle. More modern definitions express them as infinite series or solutions to specific differential equations, allowing them to extend to arbitrary positive and negative values, even complex values.

"Odd and even unchanged" means that the degree in front of the angle is a multiple of 90 degrees. If it is an even number, the name of the function does not change, and if it is an odd number, it becomes its cofunction (positive and cosine change each other, positive and cotangent change each other, and forward and cosecant change each other).

The odd orange age does not change, and the symbol looks at the quadrant.

Odd and even are unchanged (for k, k is odd or even), and the sign is seen as the quadrant (looking at the original function, and at the same time, it can be regarded as an acute angle regret).

The symbol on the right side of the equation is that when is regarded as an acute angle, the sign of the original trigonometric value of the angle k·360°+ k z ° 360°- in the quadrant can be remembered: the horizontal induced name is unchanged; symbol to see the quadrant.

How to judge the symbols of various trigonometric functions in the four quadrants, you can also remember the mantra "one is perfect;."Two sine (cosecant);Three-by-two cuts;Four cosines (secant)".

The meaning of this twelve-word mantra is to say:

The trigonometric value of any angle in the first quadrant is "+".

In the second quadrant, only the sine and cosecant are "+" and all the rest are "".

In the third quadrant, only tangent and cotangent are "+", and the cofunction of the circle is " ".

In the fourth quadrant, only secant and cosine are "+" and all the rest are "".

One is perfect, two are sine, three are tangent, and four are cosine.