Let α be an arbitrary angle, and the values of the same trigonometric functions with the same terminal edges are equal: for the positive semi-axis of X axis as the starting axis,

Representation of angles in arc system;

sin(2kπ+α)=sinα (k∈Z)

cos(2kπ+α)=cosα (k∈Z)

tan(2kπ+α)=tanα (k∈Z)

cot(2kπ+α)=cotα (k∈Z)

sec(2kπ+α)=secα (k∈Z)

csc(2kπ+α)=cscα (k∈Z)

Representation of angle in angle system;

sin (α+k 360 )=sinα(k∈Z)

cos(α+k 360 )=cosα(k∈Z)

tan (α+k 360 )=tanα(k∈Z)

cot(α+k 360 )=cotα (k∈Z)

sec(α+k 360 )=secα (k∈Z)

csc(α+k 360 )=cscα (k∈Z)[ 1]

Formula 2

Let α be an arbitrary angle, and the relationship between the trigonometric function value of π+α and the trigonometric function value of α: for the negative semi-axis of X axis as the starting axis,

Representation of angles in arc system;

Sine (π+α) =-Sine α

cos(π+α)=-cosα

tan(π+α)=tanα

cot(π+α)=cotα

sec(π+α)=-secα

csc(π+α)=-cscα

Representation of angle in angle system;

Sine (180+α)=- sine α

cos( 180 +α)=-cosα

tan( 180 +α)=tanα

cot( 180 +α)=cotα

sec( 180 +α)=-secα

CSC( 180+α)=-CSCα[ 1]

Formula 3

The relationship between arbitrary angle α and the value of-α trigonometric function;

Sine (-α) =-Sine α

cos(-α)=cosα

tan(-α)=-tanα

Kurt (-α) =-Kurt α

Second (-α) = second α

csc (-α)=-cscα[ 1]

Formula 4

The relationship between π-α and the trigonometric function value of α can be obtained by Formula 2 and Formula 3:

Representation of angles in arc system;

Sine (π-α) = Sine α

cos(π-α)=-cosα

tan(π-α)=-tanα

cot(π-α)=-coα

Seconds (π-α) =-Secondα

csc(π-α)=cscα

Representation of angle in angle system;

sin( 180 -α)=sinα

cos( 180 -α)=-cosα

tan( 180 -α)=-tanα

cot( 180 -α)=-cotα

sec( 180 -α)=-secα

csc( 180 -α)=cscα[ 1]

Formula 5

The relationship between 2π-α and the trigonometric function value of α can be obtained by formula 1 and formula 3:

Representation of angles in arc system;

Sine (2π-α)=- Sine α

cos(2π-α)=cosα

tan(2π-α)=-tanα

Kurt (2π-α)=- Kurt α

Seconds (2π-α)= Secondα

CSC(2π-α)=-csα

Representation of angle in angle system;

sin(360 -α)=-sinα

cos(360 -α)=cosα

tan(360 -α)=-tanα

cot(360 -α)=-cotα

Seconds (360-α)= Seconds α

csc(360 -α)=-cscα[ 1]

Formula 6

The relationship between π/2 α and 3 π/2 α and the trigonometric function value of α: (1 ~ 1)

Relationship between π/2+α and trigonometric function value of α

Representation of angles in arc system;

sin(π/2+α)=cosα

cos(π/2+α)=—sinα

tan(π/2+α)=-cotα

cot(π/2+α)=-tanα

sec(π/2+α)=-cscα

csc(π/2+α)=secα

Representation of angle in angle system;

sin(90 +α)=cosα

cos(90 +α)=-sinα

tan(90 +α)=-cotα

cot(90 +α)=-tanα

sec(90 +α)=-cscα

csc(90 +α)=secα[ 1]

Relationship between π/2-α and trigonometric function value of α

Representation of angles in arc system;

sin(π/2-α)=cosα

cos(π/2-α)=sinα

tan(π/2-α)=cotα

cot(π/2-α)=tanα

sec(π/2-α)= csα

csc(π/2-α)=secα

Representation of angle in angle system;

sin (90 -α)=cosα

cos (90 -α)=sinα

tan (90 -α)=cotα

cot (90 -α)=tanα

Seconds (90 -α)=cscα

csc (90 -α)=secα[ 1]

The relationship between 3 π/2+α and the trigonometric function value of α.

Representation of angles in arc system;

sin(3π/2+α)=-cosα

cos(3π/2+α)=sinα

tan(3π/2+α)=-cotα

cot(3π/2+α)=-tanα

sec(3π/2+α)= csα

csc(3π/2+α)=-secα

Representation of angle in angle system;

sin(270 +α)=-cosα

cos(270 +α)=sinα

tan(270 +α)=-cotα

cot(270 +α)=-tanα

sec(270+α)= csα

csc(270 +α)=-secα [ 1]

The relationship between 3 π/2-α and the trigonometric function value of α.

Representation of angles in arc system;

sin(3π/2-α)=-cosα

cos(3π/2-α)=-sinα

tan(3π/2-α)=cotα

cot(3π/2-α)=tanα

sec(3π/2-α)=-cscα

csc(3π/2-α)=-secα

Representation of angle in angle system;

sin(270 -α)=-cosα

cos(270 -α)=-sinα

tan(270 -α)=cotα

cot(270 -α)=tanα

Seconds (270 -α)=-cscα

csc(270 -α)=-secα[ 1]

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Inducing formula memory

Overview of inductive formula law

The function names from formula 1 to formula 5 have not changed, but the function names of formula 6 have changed.

Formulas 1 to 5 can be abbreviated as: the name of the function is unchanged, and the symbol depends on the quadrant. That is, the trigonometric function values of α+k 360 (k ∈ z), ﹣ α, 180 α, and 360-α are equal to the trigonometric function values of the same name, preceded by a sign of the original function value when α is regarded as an acute angle. [2]

The above inductive formula can be summarized as follows: the memory diagram of trigonometric formula is different from the trigonometric function value of kπ/2 α (k ∈ z),

① When k is an even number, the function value of α with the same name is obtained, that is, the function name is unchanged;

② When k is an odd number, the cofunction value corresponding to α is obtained, that is, sin→cos；; cos→sin； Tan → Kurt, Kurt → Tan. (even if it is odd, even if it is constant) and then consider α as an acute angle, add the sign of the original function value. (Symbols look at quadrants)

For example:

Sin (2π-α) = sin (4 π/2-α), and k=4 is an even number, so we take sinα.

When α is an acute angle, 2 π-α ∈ (270,360), sin (2π-α)

So sin (2 π-α) =-sin α [3]

Memory formula

Odd couples, symbols look at quadrants.

The symbols on the right side of the formula are angles k 360+α (k ∈ z),-α, 180 α, and when α is regarded as an acute angle, it is 360-α.

The sign of the original trigonometric function value in the quadrant can be remembered.

The name of horizontal induction remains unchanged; Symbols look at quadrants.

How to judge the symbols of various trigonometric functions in four quadrants, you can also remember the formula "a full pair; Two sine (cotangent); Cut in twos and threes; Four cosines (secant) ".

The meaning of this 12 formula is:

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

In the second quadrant, only sine and cotangent are "+",and the rest are "-";

In the third quadrant, only the tangent and cotangent functions are "+"and the chord function is "-";

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