"The Concept of Trigonometric Functions" Trigonometric Functions PPT (The Concept of Trigonometric Functions in Lesson 1)

Description

"The Concept of Trigonometric Functions" Trigonometric Functions PPT (The Concept of Trigonometric Functions in Lesson 1)

Part One: Learning Objectives

Understand the concept of trigonometric functions and be able to find the trigonometric function value of a given angle

Master the sign rules of trigonometric function values ​​for each quadrant angle

Master the simple application of trigonometric function induced formula 1

Concept of trigonometric functions PPT, part 2: independent learning

Problem guide

Preview textbooks P177-P181 and think about the following questions:

1. What is the definition of the trigonometric function of any angle?

2. How to determine the sign of a trigonometric function value in each quadrant?

3. What is induction formula one?

A preliminary exploration of new knowledge

1. Definition of trigonometric functions for any angle

■Instructions from famous teachers

(1) In the definition of the trigonometric function of any angle, it should be clear: α is an arbitrary angle, and its range is the set of real numbers that make the function meaningful.

(2) It is necessary to make it clear that sin α is a whole, not the product of sin and α. It is a symbol of the "sine function", just like f(x) represents a function with the independent variable x, leaving the "sin" of the independent variable "cos", "tan", etc. are meaningless.

2. Symbols of trigonometric function values

as the picture shows:

Sine: ______ quadrant is positive, ______ quadrant is negative;

Cosine: ______ quadrant is positive, ______ quadrant is negative;

Tangent: positive in the ______ quadrant, negative in the ______ quadrant.

Abbreviated formula: one perfect sine, two sines, three tangents, and four cosines.

3. Formula 1

The value of the same trigonometric function for angles with the same terminal sides is ______, thus obtaining a set of formulas (Formula 1):

sin(α+k·2π)=____________,

cos(α+k·2π)=____________,

tan(α+k·2π)=____________,

where k∈Z.

■Instructions from famous teachers

(1)The essence of formula 1

The essence of formula 1 is that for angles with the same terminal sides, the values ​​of their trigonometric functions with the same names are equal. Because the terminal sides of these angles are the same ray, according to the definition of trigonometric functions, the values ​​of the trigonometric functions of these angles are equal.

(2) The function of formula 1

Formula 1 can be used to convert the trigonometric function value of any angle into the trigonometric function value of an angle that is the same as its terminal side in the range of 0° to 360° (the method is to first find the terminal value of the given angle in the range of 0° to 360°). angles with the same sides, and then write it in the form of formula 1, and finally get the result).

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) It is known that α is the interior angle of a triangle, then sin α>0 and cos α≥0.()

(2) If sin α·cos α>0, then angle α is the first quadrant angle. ()

(3) For any angle α, the trigonometric functions sin α, cos α, and tan α are all meaningful. ()

(4) The value of the trigonometric function has nothing to do with the position of point P(x, y) on the terminal side. ()

(5) The same trigonometric function value can find countless angles corresponding to it. ()

It is known that sin α=35, cos α=-45, then the quadrant where the angle α is located is ()

A． Quadrant 1 B. second quadrant

C. The third quadrant D. Quadrant 4

It is known that the terminal side of angle α passes through P(-b, 4), and cos α=-35, then the value of b is ()

A． 3B． -3

C. ±3 D. 5

The concept of trigonometric functions PPT, the third part of the content: interactive teaching and practice

Find the value of the trigonometric function for any angle

(1) It is known that the intersection point of the terminal side of angle α and the unit circle is P35, y (y<0), find the value of tan α.

(2) It is known that the terminal edge of angle α falls on the ray y=2x (x≥0), find the values ​​of sin α and cos α.

Interactive exploration

1. (Change condition) In this example (2), the condition "the terminal side of angle α falls on ray y=2x (x≥0)" becomes "the terminal side of angle α falls on ray y=-34x (x≥0)" , find the sine, cosine and tangent of angle α.

2. (Changing conditions) In this example (2), the condition "the terminal side of α falls on the ray y=2x (x≥0)" becomes "the terminal side of α falls on the straight line y=2x", and other conditions remain unchanged. What is its conclusion?

regular method

A method to find the value of a trigonometric function when the coordinates of any point on the terminal side of α are known

(1) First use the straight line to intersect the unit circle to find the intersection coordinates, and then use the definitions of sine and cosine functions to find the corresponding trigonometric function values.

(2) Select any point P (x, y) on the terminal side of α. The distance from P to the origin is r (r>0), then sin α = yr, cos α = xr. Given the terminal side of α, find When determining the trigonometric function value of α, it is more convenient to use these formulas.

(3) When the coordinates of the point on the terminal side of angle α are given in the form of parameters, the parameters should be classified and discussed according to the actual situation of the problem.

Determination of signs of trigonometric function values

Determine the following types of symbols:

(1)tan 120°sin 269°;

(2)cos 4tan－23π4.

[Solution] (1) Because the 120° angle is the second quadrant angle,

So tan 120°<0.

Because the 269° angle is the third quadrant angle,

So sin 269°<0.

So tan 120°sin 269°>0.

Track training

1. If -π2＜α＜0, then the point (tan α, cos α) is located in ()

A． Quadrant 1B. second quadrant

C. The third quadrant D. Quadrant 4

2. (2019•The first teaching quality inspection of Anhui Taihe Middle School) It is known that sin θcos θ<0, and |cos θ|=cos θ, then the angle θ is ()

A． First quadrant angle B. second quadrant angle

C. Third quadrant angle D. Fourth quadrant angle

The concept of trigonometric functions PPT, part 4: feedback on meeting standards

1. If angle α is the third quadrant angle, then the quadrant where point P(2, sin α) is located is ()

A． Quadrant 1 B. second quadrant

C. The third quadrant D. Quadrant 4

2. If cos α=-32, and the terminal side of angle α passes through point P(x, 2), then the abscissa x of point P is ()

A． 23B. ±23 C. -22 D． -twenty three

3. cos 1 470°＝____________．

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Update Time: 2024-10-04

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