"The Concept of Trigonometric Functions" Trigonometric Functions PPT Courseware (Lesson 1: The Concept of Trigonometric Functions)

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"The Concept of Trigonometric Functions" Trigonometric Functions PPT Courseware (Lesson 1: The Concept of Trigonometric Functions)

Part One: Learning Objectives

1. Understand the definition of trigonometric functions (sine, cosine, tangent) of any angle with the help of the unit circle. (main difficulty)

2. Master the signs of any angular trigonometric function (sine, cosine, tangent) in each quadrant. (easy to make mistakes)

3. Know the formula—and apply it.

core competencies

1. Cultivate mathematical abstract literacy through the concept of trigonometric functions.

2. Improve your mathematical literacy with the help of formula operations.

PPT on the concept of trigonometric functions, part 2: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. unit circle

In the Cartesian coordinate system, we call the circle with the origin O as the center and ________ as the radius the unit circle.

2. Definition of trigonometric functions for any angle

(1)Conditions

In the plane rectangular coordinate system, assuming α is an arbitrary angle, α∈R and its terminal side intersects ______ at point P(x, y), then:

(2)Conclusion

①y is called the ______ function of α, recorded as ______, that is, sin α=y;

②x is called the ______ function of α, recorded as ______, that is, cos α=x;

③yx is called ______ of α, recorded as ______, that is, tan α=yx (x≠0).

(3) Summary

yx=tan α(x≠0) is a function with angle as the independent variable and the ratio of the ordinate or abscissa of the point on the unit circle as the function value. For the tangent function, we collectively refer to the sine function, cosine function and tangent function as trigonometric functions. function.

3. Domains of sine, cosine, and tangent functions in radians

4. The signs of sine, cosine and tangent function values ​​in each quadrant

(1)Illustration:

(2) Formula: "One is perfect, two is ______, three is ______, four is ______".

First try

1. The value of sin(-315°) is ()

A． -22B. -12

C.22D.12

2. It is known that sin α>0, cos α<0, then the angle α is ()

A． first quadrant angle

B. second quadrant angle

C． third quadrant angle

D. Fourth quadrant angle

3. sin253π=________.

4. The terminal side of angle α intersects the unit circle at point M32, 12, then the value of cos α + sin α is ________.

The concept of trigonometric functions PPT, the third part of the content: cooperative exploration to improve literacy

The definition and application of trigonometric functions

[Inquiry Questions]

1. Generally speaking, assuming that the coordinates of any point on the terminal side of angle α are (x, y), and its distance from the origin is r, what are the values ​​of sin α, cos α, and tan α?

Tip: sin α=yr, cos α=xr, tan α=yx (x≠0).

2. Do the values ​​of sin α, cos α, and tan α change with the change of the position of point P on the terminal edge?

Tip: The values ​​of sin α, cos α, and tan α are only related to the terminal edge position of α and do not change with the change of the position of point P on the terminal edge.

[Example 1] (1) It is known that there is a point P(x,3)(x≠0) on the terminal side of angle θ, and cos θ=1010x, then the value of sin θ+tan θ is ________.

(2) It is known that the terminal side of angle α falls on the straight line 3x+y=0, find the values ​​of sin α, cos α and tan α.

[Ideas Enlightenment] (1) Find x based on the definition of a series of equations based on the cosine function→

Find sin θ＋tan θ according to the definition of sine and tangent functions

(2) Determine the terminal edge position of angle α → Classify and discuss to find sin α, cos α, tan α

regular method

Steps to find the value of the trigonometric function from the coordinates of any point on the terminal side of angle α:

(1) When the terminal side of the known angle α is on a straight line, there are two commonly used problem-solving methods:

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. When the terminal side of α is known, it is more convenient to use these formulas to find the trigonometric function of α.

(2) When the coordinates of the point on the terminal side of the angle α are given in parameter form, attention must be paid to distinguishing the positive and negative letters. If the positive and negative are not determined, they need to be classified and discussed.

Use of Trigonometric Function Value Symbols

[Example 2] (1) It is known that point P (tan α, cos α) is in the fourth quadrant, then the terminal side of angle α is in ()

A． Quadrant 1 B. second quadrant

C． The third quadrant D. Quadrant 4

(2) Determine the following symbols:

①sin 145°cos(-210°); ②sin 3•cos 4•tan 5.

[Idea Tips] (1) First determine the signs of tan α and cos α, and then determine in which quadrant the terminal side of angle α is.

(2) First determine which quadrant angle the known angle is, then determine the sign of each trigonometric function value, and finally determine the sign of the product.

regular method

Strategies for determining the signs of trigonometric function values ​​in each quadrant:

1 Basics: Accurately determine the quadrant of each angle in the trigonometric function value;

2 Key: Accurately memorize the symbols of trigonometric functions in each quadrant;

3 Note: Angle given in radian system often does not write the unit. Do not mistakenly think that the angle will lead to wrong judgment of the quadrant.

Reminder: Pay attention to the clever use of formulas to memorize the symbols of trigonometric function values ​​in each quadrant.

Class summary

1. Learning the definition of trigonometric functions is the basis for learning all trigonometric function knowledge in the future. It is necessary to fully understand its connotation and grasp the key point that the value of a trigonometric function is only related to the position of the terminal side of the angle and has nothing to do with the selected point.

2. The first induction formula refers to the fact that trigonometric functions with the same name and terminal sides having the same angle are equal in value, but the converse is not necessarily true. When memorizing, you can combine it with the definition of trigonometric functions.

3. The sign of the trigonometric function value in each quadrant mainly involves the square root and the calculation of the absolute value. At the same time, attention should also be paid to the sign of the sine and cosine of the terminal side on the coordinate axis.

PPT on the concept of trigonometric functions, part 4: Achieve standards in class and solidify double bases

1. Thinking and analysis

(1) sin α represents the product of sin and α. ()

(2) Assume the point P(x, y) on the terminal side of angle α, r=|OP|≠0, then sin α=yr, and the larger y is, the larger the value of sin α is. ()

(3) The values ​​of the same trigonometric function of angles with the same terminal sides are equal. ()

(4) The value of the tangent function of the angle where the terminal side falls on the y-axis is 0.()

[Tips] (1) Error. sin α represents the sine value of angle α, which is a "whole".

(2)Error. From the definition of the sine function of any angle, sin α=yr. But when y changes, sin α is a constant value.

(3) Correct.

(4)Error. The value of the tangent function of an angle whose terminal side falls on the y-axis does not exist.

2. It is known that the terminal side of angle α passes through point P(1,-1), then the value of tan α is ()

A． 1

B. -1

C.22

D. -twenty two

3. In the plane rectangular coordinate system xOy, angle α and angle β both have Ox as the starting side, and their terminal sides are symmetrical about the x-axis. If sin α=15, then sin β=________.

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Update Time: 2024-11-22

This template belongs to Mathematics courseware People's Education High School Mathematics Edition A Compulsory Course 1 industry PPT template

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