"Trigonometric Identity Transformation" Trigonometric Functions PPT Courseware (Cosine Formula of the Difference of Two Angles in Lesson 1)

Description

"Trigonometric Identity Transformation" Trigonometric Functions PPT Courseware (Cosine Formula of the Difference of Two Angles in Lesson 1)

Part One: Learning Objectives

1. Understand the derivation process of the cosine formula of the difference between two angles. (emphasis)

2. Understand the main steps in deriving formulas using the vector method. (difficulty)

3. Memorize the form and symbolic characteristics of the cosine formula of the difference between two angles, and be able to use this formula for evaluation and calculation. (Key point, easy to confuse)

core competencies

1. Cultivate mathematical operation literacy through the derivation of the cosine formula of the difference between two angles.

2. Use the transformation, positive and reverse uses of formulas to improve your logical reasoning skills.

Trigonometric Identity Transformation PPT, Part 2: Independent preview to explore new knowledge

A preliminary exploration of new knowledge

Cosine formula for the difference between two angles

Formula cos(α-β)＝

Applicable conditions: The angles α and β in the formula are arbitrary angles

Formula structure: The two parts on the right side of the formula are the products of trigonometric functions of the same name, and the connecting sign is opposite to the connecting sign on the left corner.

First try

1. sin 14°cos 16°+sin 76°cos 74°＝()

A． 32B. 12

C． -32 D． -12

2. The value of cos(-15°) is ()

A.6－22 B.6＋22

C.6－24 D.6＋24

3. cos 65°cos 20°+sin 65°sin 20°＝________.

Trigonometric Identity Transformation PPT, Part 3: Collaborative exploration to improve literacy

Angle evaluation problem

[Example 1] (1) The value of cos13π12 is ()

A.6＋24　B.6－24

C.2-64 D． -6+24]

(2) Find the values ​​of the following expressions:

①cos 75°cos 15°－sin 75°sin 195°;

②sin 46°cos 14°+sin 44°cos 76°;

③12cos 15°+32sin 15°.

regular method

1. The general idea for solving the evaluation problem of trigonometric function expressions containing non-special angles is:

(1) Convert non-special angles into the sum or difference of special angles and directly evaluate them using formulas.

(2) In the conversion process, make full use of the induction formula to construct the structural form of the cosine formula of the difference between two angles, and then use the formula inversely to evaluate it.

2. The structural features of the cosine formula of the difference between two angles:

(1) Multiplication of functions with the same name: that is, the cosine of two angles multiplied by the cosine, and the sine multiplied by the sine.

(2) Add the products obtained.

Track training

1. Simplify the following formulas:

(1)cos(θ+21°)cos(θ-24°)+sin(θ+21°)sin(θ-24°);

(2)－sin 167°•sin 223°+sin 257°•sin 313°.

Problems with evaluating values ​​(formulas)

[Inquiry Questions]

1. If the trigonometric function values ​​of α + β and β are known, how to find the value of cos α?

Tip: cos α=cos[(α+β)-β]

=cos(α+β)cos β+sin(α+β)sin β.

2. Using α-(α-β)=β, we can get what cos β is equal to?

Tip: cos β=cos[α-(α-β)]=cos αcos(α-β)+sin αsin(α-β)．

regular method

Problem-solving strategies for value-based evaluation problems

1. When the values ​​of trigonometric functions of certain angles are known, and when finding the values ​​of trigonometric functions of other angles, attention should be paid to the relationship between the known angles and the angles in the expression being sought, that is, the angles are divided and the angles are rounded up.

2 Since the sum and difference angles are relative to single angles, the angles can be flexibly removed or combined as needed during the problem-solving process. Common angle transformations include:

①α＝α－β＋β;

②α=α+β2+α-β2;

③2α＝α＋β＋α－β;

④2β＝α＋β－α－β.

Class summary

1. The problem of evaluating given expressions or evaluating given values ​​is to find the trigonometric function values ​​of other angles from given certain functional expressions or the trigonometric function values ​​of certain angles. The key lies in "variation" or "variation of angles" , replace "target angle" with "known angle". Pay attention to the formal use, reverse use, and deformation of formulas. Sometimes it is necessary to use techniques such as splitting corners and spelling corners.

2. The problem of "finding an angle by giving a value" can actually be transformed into a problem of "finding an angle by giving a value". To find the value of an angle, it can be divided into the following three steps: 1. Find the value of a certain trigonometric function of the angle; 2. Determine the location of the angle. Range (find a monotonic interval); ③ Determine the value of the angle. Determining which trigonometric function value to use for the angle sought depends on the specific question.

Trigonometric Identity Transformation PPT, Part 4: Complying with Standards and Solidifying the Double Basics in the Hall

1. Thinking and analysis

(1)cos(60°-30°)＝cos 60°-cos 30°.()

(2) For any real numbers α, β, cos(α-β)=cos α-cos β is not true. ()

(3) For any α, β∈R, cos(α-β)=cos αcos β＋sin αsin β is true. ()

(4)cos 30°cos 120°+sin 30°sin 120°＝0.()

[Tips] (1) Error. cos(60°-30°)＝cos 30°≠cos 60°-cos 30°.

(2)Error. When α=-45°, β=45°, cos(α-β)=cos(-45°-45°)=cos(-90°)=0, cos α-cos β=cos(-45° )-cos 45°=0, at this time cos(α-β)=cos α-cos β.

(3) Correct. The conclusion is the cosine formula of the difference between two angles.

(4) Correct. cos 30°cos 120°＋sin 30°sin 120°＝cos(120°－30°)＝cos 90°＝0.

2. It is known that α is an acute angle, β is the third quadrant angle, and cos α=1213, sin β=-35, then the value of cos(α-β) is ()

A． -6365B． -3365

C.6365　D.3365

3. cos(α-35°)cos(α+25°)+sin(α-35°)sin(α+25°)=________.

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Update Time: 2024-10-31

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