"Inducing Formulas" Trigonometric Functions PPT Courseware (Inducing Formulas 2, 3, and 4 in Lesson 1)

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"Inducing Formulas" Trigonometric Functions PPT Courseware (Inducing Formulas 2, 3, and 4 in Lesson 1)

Part One: Learning Objectives

1. Understand the derivation methods of Formula 2, Formula 3 and Formula 4.

2. Able to accurately memorize Formula 2, Formula 3 and Formula 4. (Key point, easy to confuse)

3. Master formula 2, formula 3 and formula 4, and be able to apply them flexibly. (difficulty)

core competencies

1. Use formulas to perform calculations and cultivate mathematical literacy.

2. Improve logical reasoning skills through simplification and proof through deformation of formulas.

Induction formula PPT, the second part: independent prediction and exploration of new knowledge

A preliminary exploration of new knowledge

1. Formula 2

(1) The terminal sides of angle π+α and angle α are symmetrical about _____. as the picture shows.

(2)Formula: sin(π+α)=_____,

cos(π+α)=_____,

tan(π+α)=_____.

2. Formula 3

(1) The terminal sides of angle -α and angle α are symmetrical about the axis. as the picture shows.

(2)Formula: sin(-α)=_____,

cos(-α)=_____,

tan(-α)=_____.

3. Formula 4

(1) The terminal sides of angle π-α and angle α are symmetrical about the axis. as the picture shows.

(2)Formula: sin(π-α)=_____,

cos(π-α)=_____,

tan(π-α)=_____.

Thinking: (1) Can the angle α in the induction formula only be an acute angle?

(2) Do induced formulas 1 to 4 change the name of the function?

Tips: (1) The angle α in the induction formula can be any angle. Please note that the tangent function requires α≠kπ+π2, k∈Z.

(2) The induction formulas 1 to 4 do not change the function name.

First try

1. If α and β satisfy α+β=π, then the correct number in the following formula is ()

①sin α=sin β; ②sin α=-sin β; ③cos α=-cos β; ④cos α=cos β; ⑤tan α=-tan β.

A． 1B. 2C. 3D. 4

C　Because α+β=π, so sin α=sin(π-β)=sin β,

Therefore, ① is correct and ② is wrong;

cos α=cos(π-β)=-cos β,

Therefore, ③ is correct and ④ is wrong;

tan α=tan(π-β)=-tan β, ⑤ is correct.

So choose C.]

2. tan-4π3 is equal to ()

A． -33 B.33

C． -3D.3

3. It is known that tan α=3, then tan(π+α)=________.

4. Evaluate: (1)sin2π3=________.

(2)cos-7π6=________.

Induction formula PPT, the third part: cooperative exploration to improve literacy

Angle evaluation problem

[Example 1] Find the values ​​of the following trigonometric functions:

(1) sin 1 320°; (2) cos-31π6; (3) tan (-945°).

[ Solution

Method 2: sin 1 320°＝sin(4×360°-120°)＝sin(-120°)

=-sin(180°-60°)=-sin 60°=-32.

(2) Method 1: cos-31π6=cos31π6

=cos4π+7π6=cosπ+π6=-cosπ6=-32.

Method 2: cos-31π6=cos-6π+5π6

=cosπ-π6=-cosπ6=-32.

(3)tan(-945°)＝-tan 945°＝-tan(225°+2×360°)

=-tan 225°＝-tan(180°+45°)＝-tan 45°＝-1.

regular method

Steps to use the induction formula to find the value of the trigonometric function of any angle

1 "Convert negative to positive" - ​​use formula 1 or 3 to transform;

2. "Making it larger" - use formula 1 to change the angle into an angle between 0° and 360°;

3. "Minimize and sharpen" - use formula 2 or 4 to convert angles greater than 90° into acute angles;

4 "Sharp evaluation" - evaluate the trigonometric function after obtaining the acute angle.

Class summary

1. The induction formulas 1 to 4 can be briefly summarized as "α+k·2π(k∈Z), -α, the trigonometric function value of π±α is equal to the function value of the same name of α, plus the original function value when α is regarded as an acute angle. symbol". Or simply described as "the function has the same name and the quadrant number".

2. Using formulas 1 to 4, the trigonometric function of any angle can be converted into an acute angle trigonometric function. Generally, the following steps can be followed:

The trigonometric function of any negative angle ��→use formula 3 or the trigonometric function of any positive angle ��→use formula 1

Trigonometric functions of angles from 0 to 2π → Use formula 2 or trigonometric functions of four acute angles

Inducement formula PPT, the fourth part: reaching the standard in class and solidifying the base

1. Thinking and analysis

(1) Formulas 2 to 4 are valid for any angle α. ()

(2) From formula three, we know cos-(α-β)=-cos(α-β). ()

(3) In △ABC, sin(A+B)=sin C. ()

[Tips] (1) Error, among the three formulas about tangent, α≠kπ+π2, k∈Z.

(2) From formula three, we know cos-(α-β)=cos(α-β),

Therefore, cos[-(α-β)]=-cos(α-β) is incorrect.

(3) Because A+B+C=π, so A+B=π-C,

So sin(A+B)=sin(π-C)=sin C.

2. It is known that sin(π+α)=35, and α is the fourth quadrant angle, then the value of cos(α-π) is ()

A.45B. -45

C． ±45　D.35

3. The value of cos-585°sin 495°+sin-570° is equal to ________.

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Update Time: 2024-10-04

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

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