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

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

Part One: Learning Objectives

Understand half-angle and its derivation process

Flexibly use the sine and cosine formulas of sum and difference to perform relevant calculations, simplifications and proofs

Trigonometric Identity Transformation PPT, Part 2: Independent Learning

Problem guide

Preview textbooks P225-P228 and think about the following questions:

1. How to express sin2α2, cos2α2 and tan2α2 in terms of cos α?

2. What factors determine the sign of a half-width formula?

A preliminary exploration of new knowledge

1. half angle formula

2. Auxiliary angle formula

asin x + bcos x = a2 + b2 sin (x + θ) (where tan θ = ba).

self-test

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

(1) The half-angle formula is applicable to any angle. ()

(2)cos α2＝ 1＋cos α2.()

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

If cos α=13, and α∈(0, π), then the value of cos α2 is ()

A． 63B. -63

C． ±63 D. ±33

It is known that cos α=45, α∈3π2, 2π, then sin α2 is equal to ()

A． -1010B. 1010

C． 3310 D. -35

It is known that cos θ=-35, and 180°<θ<270°, then tan θ2=________.

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

Use the half-angle formula to evaluate

It is known that α is an obtuse angle, β is an acute angle, sin α=45, sin β=1213, find the value of cos α-β2.

regular method

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

(1) Look at the angle: If the angle in the trigonometric function formula is known to be twice the angle in the trigonometric function formula to be solved, the half-angle formula is often used to solve the problem.

(2) Clear range: Since the evaluation of half-angle formulas often involves sign problems, it is necessary to find the range of the corresponding half-angle based on the range of the angle when solving.

(3) Select formula: When it comes to the tangent value of the half-angle formula, tan α2=sin α1+cos α=1-cos αsin α is commonly used. The advantage is that it can avoid the angle range problem caused by the square root during calculation; when it comes to the half-angle formula When calculating the sine and cosine values, sin2 α2=1-cos α2 and cos2 α2=1+cos α2 are often used to calculate.

Simplification of trigonometric functions

Simplify (1-sin α-cos α) sin α2+cos α22-2cos α(-π<α<0).

regular method

Ideas and methods for simplifying trigonometric functions

(1) Ideas for simplification: For sum formulas, the basic idea is to reduce degrees, eliminate terms and inverse formulas; for trigonometric formulas, the basic idea is to reduce the numerator and denominator or use inverse formulas; for quadratic radical formulas, pay attention to the double angle Converse application of formula. In addition, methods such as tangential chords, variable substitution, and angle normalization can also be used.

(2) Methods of simplification: convert strings to each other, convert different names into the same name, convert different angles into the same angle, lower or raise powers, etc.

Trigonometric Identity Transformation PPT, Part 4: Feedback on Compliance

1. If sin(π-α)=-53 and α∈π, 3π2, then sinπ2+α2 is equal to ()

A． -63B． -66

C． 66 D. 63

2. Simplify: 1+cos(3π-θ)23π2<θ<2π=________.

3. It is known that α∈0, π2, β∈π2, π, cos β=-13, sin(α+β)=79.

(1) Find the value of tanβ2;

(2) Find the value of sin α.

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