Thinking Guide of Trigonometric Function (Middle)-1

One: Overview

In the last section, we introduced the angle system and arc system of trigonometric function and their basic properties. Next, the basic relations and inductive formulas in trigonometric function identity transformation are introduced. Figure 1 is still a mind map for us to learn trigonometric functions.

Two: identity conversion

The trigonometric function identity transformation is not only often used to simplify, evaluate and prove trigonometric identities, but also because some algebraic problems can be transformed into trigonometric problems through trigonometric substitution. Many positional relationships in solid geometry are described by their intersection angles, and finally reflected by trigonometric problems. Due to the establishment of parametric equation, the curve problem in analytic geometry can be reduced to a triangle problem. Therefore, trigonometric identity transformation involves a wide range in the whole high school mathematics and is a common "tool" for solving problems. Trigonometric function identity transformation is widely used in high school mathematics. It is very necessary to have a "global map" in mind before mastering the trigonometric function identity transformation. Fig. 2 is a mind map of trigonometric function identity transformation.

2. 1 basic relationship

2. The square relation of1.1trigonometric function.

2. 1. 1. 1 The first one is (Sina) 2+(COSA) 2 = 1. This is easy to remember and the derivation process is simple. We now deduce the process of this square relationship. Figure 3 is a right triangle, and the hypotenuse c is 1.

Because: Sina = A/C, COSA = B/C.

Also: A 2+B 2 = C 2

So (Sina) 2+(COSA) 2

=(a/c)^2+(b/c)^2

=(a^2+b^2)/c^2

=c^2/c^2

= 1

By remembering Pythagorean theorem, we can simply and quickly deduce Tao (Sina) 2+(COSA) 2 = 1.

2. 1. 1.2 The second one is 1+(tana) 2 = (seca) 2. We still use Pythagorean theorem to derive this formula.

Because: SECA = C/B, Tana = A/B.

Again: C 2-A 2 = B 2

So: (Seka) 2- (Tana) 2

? =(c/b)^2-(a/b)^2

? =(c^2-a^2)/b^2

? =b^2/b^2

? = 1

Similarly, if we remember Pythagorean theorem, we can simply and quickly deduce Tao 1+(tana) 2 = (seca) 2.

2. 1. 1.3 The third one is 1+(COTA) 2 = (CSCA) 2. Everything else is the same, it's still this triangle, or the formula is deduced by Pythagorean theorem.

Because: CSCA = C/A, COTA = B/A.

Again: c 2-b 2 = a 2

So: (CSCA) 2-(COTA) 2

=(c/a)^2-(b/a)^2

=(c^2-b^2)/a^2

=a^2/b^2

? = 1。

2. 1. 1.4 To sum up, the square relation of trigonometric functions is nothing more than the pythagorean theorem.

2. Quotient relation of1.2 trigonometric function.

2. 1.2. 1 The first one is tanA = sinA/cosA. This is easy to deduce, as follows.

Because: Sina = a/c, COSA = b/c;

TanA = a/b

So: Sina /cosA

= (AC)/(DC)

=a/b

= Tana

2. 1.2.2 The second one is cotA = cosA/sinA. This is also easy to deduce, as follows.

Because: Sina = a/c, COSA = b/c;

Also: cotA = b/a

So: cosA/ Sina

=(b/c)/(a/c)

=b/a

=cotA

2. Reciprocal relation of1.3 trigonometric function.

2. 1.3. 1 The first one is Sina *cscA =

1。 This is easy to deduce, as follows.

Because: Sina = a/c, CSCA = c/a;

So: Sina *cscA

= (communication) * (communication)

= 1

2. 1.3.2 The second one is cosA*secA =

1。 This is easy to deduce, as follows.

Because: cosA = b/c, SECA = c/b;

So: cosA*secA

=(b/c)*(c/b)

= 1

2. 1.3.3 the third one is tana*cota =

1。 This is easy to deduce, as follows.

Because: tanA = a/b, cota = b/a;

So: Tana Kota

=(a/b)*(b/a)

= 1

2. 1.4 Summary of basic relations of trigonometric functions. The so-called square relation is another form of Pythagorean theorem in trigonometric functions. The quotient relation of trigonometric functions is nothing more than the proportional relation of each side of a right triangle. The reciprocal relationship of trigonometric functions is the same. We can also use the chart in Figure 4 to understand the relationship between them more intuitively.

2.2 inductive formula

2.2. 1 All formulas exist to solve complex problems more easily. Now I will introduce the function of trigonometric function induction formula to you: it is to transform trigonometric function with arbitrary angle into trigonometric function with acute angle. Give a simple example.

sin 390 = sin(360+30)= sin 30 = 1/2。

tan 225 = tan( 180+45)= tan 45 = 1。

cos 150 = cos(90+60)= sin 60 =√3/2。

Predecessors summed up a sentence, "Odd variables remain unchanged, symbols look at quadrants", so the inductive formula can be used simply and conveniently. How to understand these eight words?

Induction angles: 0,90,180,270,360. The "parity constant" is aimed at these five induction angles.

90 and 270 are 1 times, which is three times that of 90, so they are "odd"; 0, 180 and 360 are 0, 2 and 4 times of 90, so they are even numbers. 90 α and 270 α should be "changed"; 0 α,180 α, 360 α, all "unchanged". Change what? How to change it? What changes is the name of the function, and the method is positive complementary mutual change: sine changes to cosine, and cosine changes to sine; Tangent becomes cotangent, cotangent becomes tangent; Secant becomes cotangent and cotangent becomes secant.

Symbol depends on quadrant: when using induction formula, remember that no matter how big α is behind the induction angle, it should be regarded as an "acute angle", so as to decide which quadrant symbol to use. For example, sin (90+500) = cos 500 and the induction angle is 90, then sin becomes cos. If 500 is regarded as an acute angle, then 90+500 is regarded as the angle of the second quadrant, and sin is positive, so it is still positive after becoming cos. Another example is tan (180-425) =-tan 425, because the induced angle is 180, which is an "even constant" and 425 is regarded as an acute angle, so 180-425 is the angle of the second quadrant (-360).

If you understand the above laws and principles, you can choose the sensing angle at will. For example, your example is: sin( 17π/2-α)=cosα.

This is because 17(π/2) is 17 times of 90, which is "odd". If sin is replaced by cos, 17π/2-α is regarded as 90-α in the first quadrant, and sin in the first quadrant is positive, so the positive sign is taken before cos. sin( 18π/2-α)= sin(9π-α)= sinα。 This is because 18(π/2) is an even multiple of 90, "unchanged", so it is still SIN, and the symbol takes the symbol of the second quadrant SIN.

At present, it is still safe. The trigonometric function with too large an angle is changed from 360 α to a trigonometric function with a smaller angle, and then the inductive formula is changed to an acute trigonometric function. For example:

sin( 17π/2-α)= sin(8π+π/2-α)= sin(π/2-α)= cosα；

sin( 18π/2-α)= sin(9π-α)= sin(8π+π-α)= sin(π-α)= sinα。

The induced angles here are all 8π, which is four times that of 2π. The name of the function remains the same, and the symbols are all in the first quadrant, because π/2-α and.

π-α should be regarded as acute angle.

The following is the specific formula of the inductive formula.

Formula 1: Let α be any angle, and the values of the same trigonometric function with the same terminal edge are equal.

sin(2kπ+α)=sinα(k∈Z)

cos(2kπ+α)=cosα(k∈Z)

tan(2kπ+α)=tanα(k∈Z)

cot(2kπ+α)=cotα(k∈Z)

Equation 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.

Sine (π+α) =-Sine α

cos(π+α)=-cosα

tan(π+α)=tanα

cot(π+α)=cotα

Equation 3: Relationship between trigonometric function values of arbitrary angles α and-α.

Sine (-α) =-Sine α

cos(-α)=cosα

tan(-α)=-tanα

Kurt (-α) =-Kurt α

Formula 4: The relationship between π-α and the trigonometric function value of α can be obtained by using Formula 2 and Formula 3.

Sine (π-α) = Sine α

cos(π-α)=-cosα