"Trigonometric Identity Transformation" Trigonometric Functions PPT Courseware (Simple Trigonometric Identity Transformation in Lesson 5)

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"Trigonometric Identity Transformation" Trigonometric Functions PPT Courseware (Simple Trigonometric Identity Transformation in Lesson 5)

Part One: Learning Objectives

1. Can use the double angle formula to derive the half-angle formula, and can use the trigonometric function formula of the sum and difference of two angles to derive the product sum and difference, and sum and difference product formulas. Understand the basic thinking and methods of trigonometric identity transformation and make simple applications. (emphasis)

2. Understand the characteristics and transformation techniques of trigonometric identity transformations, master the basic thinking and methods of trigonometric identity transformations, and be able to use trigonometric identity transformations to simplify and evaluate trigonometric function expressions, as well as prove trigonometric identities and some simple applications. (Difficult and error-prone points)

core competencies

1. Cultivate logical reasoning skills through the derivation of formulas.

2. Improve mathematical operation literacy with the help of simple applications of trigonometric identity transformation.

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

half angle formula

(1) sinα2＝± 1-cos α2,

(2) cos α2 = ± 1 + cos α2,

(3)tanα2＝± 1-cos α1＋cos α,

(4)tanα2＝sin α2cosα2＝sinα2·2cosα2cosα2·2cosα2＝sin α1＋cos α,

tanα2＝sinα2cosα2＝sinα2·2sinα2cosα2·2sinα2＝1－cos αsin α.

First try

1. It is known that 180°＜α＜360°, then the value of cosα2 is equal to ()

A． -1-cos α2　B.1-cos α2

C． -1＋cos α2 D.1＋cos α2

2. It is known that cos α=35, α∈3π2, 2π, then sin α2 is equal to ()

A.55B. -55

C.45D.255

3. It is known that 2π＜θ＜4π, and sin θ=-35, cos θ＜0, then the value of tanθ2 is equal to ________.

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

Simplify the evaluation problem

[Example 1] (1) Suppose 5π＜θ＜6π, cosθ2=a, then sinθ4 is equal to ()

A.1＋a2　B.1－a2

C． -1＋a2 D． -1-a2

(2) It is known that π<α<3π2, simplify:

1+sin α1+cos α-1-cos α+1-sin α1+cos α+1-cos α.

[Idea Tips] (1) First determine the range of θ4, and then evaluate it according to the formula of sin2θ4=1-cosθ22.

(2) 1+cos θ=2cos2α2, 1-cos α=2sin2α2, remove the root sign, determine the range of α2, and simplify.

regular method

1. Simplify the “three changes” in the problem

(1) Angle change: When performing trigonometric transformation, we usually first look for the connection between the angles in the formula, eliminate the differences between the angles by splitting and joining together, and reasonably choose the formula that connects them.

(2) Name changes: Observe the differences in types of trigonometric functions and try to unify the names of the functions, such as unifying them as chords or unifying them as tangents.

(3) Variation: Observe the difference in the structural form of the formula and choose the appropriate transformation method, such as raising power, lowering power, formula, square root, etc.

2. The idea of ​​using half-angle formula to evaluate

(1) Look at the angle: look at the 2 times relationship between the known angle and the angle to be found.

(2) Clear range: Find the range of the corresponding half-angle to prepare for determining the symbol.

(3) Select formula: When the tangent value of the half-angle formula is involved, tanα2=sin α1+cos α=1-cos αsin α is often used. When the sine and cosine values ​​of the half-angle formula are involved, sin2α2=1-cos α2, cos2α2=1+cos α2 calculate.

(4) Draw a conclusion: combine (2) to evaluate.

Reminder: Knowing the value of cos α, you can find the sine, cosine, and tangent values ​​of α2. Pay attention to determine their signs.

Proof of Trigonometric Identities

[Example 2] Verify: cos2α1tanα2-tanα2=14sin 2α.

[Thinking Points] Method 1: Use the double angle formula to prove the tangent chord from left to right;

Method 2: Keep cos2α unchanged, directly use the double angle tangent formula to deform.

regular method

Common methods for proving trigonometric identities

1. The method of determining cause and effect: the form of proof is generally simplified from complex to complex;

2. Left and right normalization method: prove that both left and right sides are equal to the same formula;

3. Patchwork method: Aiming at the differences between the question and the conclusion, targeted deformation is used to eliminate the differences between them. In short, it is to resolve differences and seek common ground;

4 Comparison method: try to prove "left - right = 0" or "left/right = 1";

5 Analysis method: Starting from the proved equation, gradually explore the conditions that make the equation true, until the conditions or obvious facts are known, you can conclude that the original equation is true.

Application of trigonometric functions in practical problems

[Inquiry Questions]

1. When using trigonometric functions to solve practical problems, what are usually chosen as independent variables? What should we pay attention to when finding the domain?

Tip: Usually the angle is used as the independent variable. When finding the domain, attention should be paid to the actual meaning and the influence of the boundedness of the sine and cosine functions.

2. After establishing a trigonometric function model, what form does the function usually need to be analytically expressed in?

Tip: Change it into the form y=Asin(ωx+φ)+b.

regular method

Methods and precautions for applying trigonometric functions to solve practical problems

1 Method: To answer this kind of problem, the key is to reasonably introduce auxiliary angles, determine the relationship between various quantities, convert the actual problem into a trigonometric function problem, and then use the relevant knowledge of trigonometric functions to solve the problem.

2 Note: During the solution process, three points should be paid attention to: ① Make full use of the geometric properties of the plane to find quantitative relationships. ② Pay attention to the range of variables in practical problems. ③ Pay attention to the influence of the boundedness of trigonometric functions.

Reminder: When using trigonometric transformation to solve practical problems, errors are often caused by ignoring the range of angles.

Class summary

1. When learning trigonometric identity transformations, don't just memorize formulas and ignore the understanding of thinking methods. You must learn to use the previous limited formulas to deduce subsequent formulas, based on memorizing formulas and applying formulas in the process of formula derivation.

2. To study the properties of functions of the form f(x)=asin x+bcos x, we must use the auxiliary angle to formulate it into the form of a sine function or cosine function of an overall angle. Therefore, the auxiliary angle formula is an important formula that is widely used in trigonometric functions and is also one of the test points of the college entrance examination. Some special coefficients a and b should be mastered proficiently. For example, sin x±cos x=2sinx±π4; sin x±3cos x=2sinx±π3, etc.

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

1. Thinking and analysis

(1)cos α2＝1＋cos α2.()

(2) There exists α∈R such that cos α2=12cos α.()

(3) For any α∈R, sin α2=12sin α is not true. ()

(4) If α is the first quadrant angle, then tan α2=1-cos α1+cos α.()

2. If f(x)=cos x-sin x is at [0, a] is a decreasing function, then the maximum value of a is ()

A.π4B.π2

C.3π4D. π

3. The minimum positive period of the function f(x)=sin2x is _________.

4. The logo of the International Congress of Mathematicians held in Beijing is based on the chord diagram of ancient Chinese mathematician Zhao Shuang. A chord diagram is composed of four congruent right triangles and a small square to form a large square (as shown in the picture). If the area of ​​the small square is 1, the area of ​​the large square is 25, and the smaller acute angle in the right triangle is θ, find cos 2θ.

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Update Time: 2024-11-03

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