"Trigonometric Identity Transformation" Trigonometric Functions PPT (Forms of sine, cosine and tangent of double angles in Lesson 4)

Description

"Trigonometric Identity Transformation" Trigonometric Functions PPT (Forms of sine, cosine and tangent of double angles in Lesson 4)

Part One: Learning Objectives

Be able to derive the sine, cosine, and tangent formulas for double angles

Be able to flexibly use the double angle formula to solve problems such as evaluation, simplification and proof.

Trigonometric Identity Transformation PPT, Part 2: Independent Learning

Problem guide

Preview textbooks P220-P223 and think about the following questions:

1. In the formulas C(α+β), S(α+β) and T(α+β), if α=β, are the formulas still valid?

2. In the above formula, if α = β, what conclusion can be drawn?

A preliminary exploration of new knowledge

Formulas for sine, cosine, and tangent of double angles

■Instructions from famous teachers

Correctly understand the double angle formula

(1) It should be noted that the prerequisite for the application of the formula is that the trigonometric functions included are meaningful.

(2) The "multiple angle" in the angle multiple formula is relative and holds true for any situation where the ratio of two angles is equal to 2. For example, 4α is twice as much as 2α, and α is twice as much as α2. This contains the idea of ​​metamorphosis. That is to say, "times" is a relative term and describes the relationship between two quantities.

self-test

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

(1)10α is a multiple of the angle of 5α, and 5α is a multiple of the angle of 5α2. ()

(2) The applicable range of the sine, cosine and tangent formulas of double angles is any angle. ()

(3) There is an angle α such that sin 2α = 2sin α holds. ()

(4) For any angle α, there is always tan 2α=2tan α1-tan2α.()

It is known that sin α=35, cos α=45, then sin 2α is equal to ()

A.75B.125

C.1225 D.2425

The result of calculating 1-2sin222.5° is equal to ()

A.12 B.22

C.33 D.32

Trigonometric Identity Transformation PPT, Part 3: Lecture and Practice Interaction

Evaluate the angle

Find the values ​​of the following expressions.

(1)sinπ8cosπ8;

(2) cos2π6-sin2π6;

(3)2tan 150°1-tan2150°;

(4)cos π5cos 2π5.

regular method

Two types of solutions to angle evaluation problems

(1) Directly use the double angle formula directly and inversely, and combine the induction formula and the basic relationship of the trigonometric functions of the same angle to transform the known formula, which can generally be converted into a special angle.

(2) If the form is the multiplication of trigonometric functions of several non-special angles, the sine formula of twice the angle is generally used inversely. During the solution process, the conditions for applying the double angle formula need to be matched by using the mutual complement relationship, so that the problem There is a form that can be used together with the sine formula of twice the angle.

Evaluate a value

It is known that π2<α<π, sin α=45.

(1) Find the value of tan 2α;

(2) Find the value of cos2α-π4.

Solving strategy

General ideas on trigonometric function evaluation problems

(1) One is to deform the conditions of the question, moving the angles and function names in the conditions closer to the angles and function names in the conclusion; the other is to deform the conclusion, moving the names of angles and functions in the conclusion closer to the names of the questions The corners and function names in the conditions should be brought closer together so that the conditions of the question can be substituted into the conclusion.

(2) Pay attention to the flexible application of several formulas, such as:

①sin 2x＝cosπ2－2x＝cos2π4－x

=2cos2π4-x-1=1-2sin2π4-x;

②cos 2x＝sinπ2－2x＝sin2π4－x

=2sinπ4-xcosπ4-x.

regular method

Simplification and proof of trigonometric function expressions

(1) Simplification method

① Chords are transformed into each other, different names are transformed into the same name, and different angles are transformed into the same angle; ② Descending or raising power; ③ An important conclusion: (sin θ±cos θ)2=1±sin 2θ.

(2) Method of proving trigonometric identities

① Start with the complicated side and prove that one side is equal to the other side; ② Comparison method, left side - right side = 0, left side and right side = 1; ③ Analysis method, start from the equation to be proved, and find the conditions for the establishment of the equation step by step.

Trigonometric Identity Transformation PPT, Part 4: Feedback on Compliance

1. It is known that sin α=3cos α, then the value of tan 2α is ()

A． 2B． -2

C． 34D． -34

2. It is known that sin θ2 + cos θ2 = 233, then sin θ = _____, cos 2θ = ______.

3.cos π12－sin π12cos π12＋sin The value of π12 is ________.

4. It is known that α∈π2, π, sin α=55.

(1) Find the values ​​of sin 2α and cos 2α;

(2) Find the value of cos5π6-2α.

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Update Time: 2024-11-18

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