"The Concept of Trigonometric Functions" Trigonometric Functions PPT Courseware (Basic Relationships of Trigonometric Functions with Same Angles in Lesson 2)

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"The Concept of Trigonometric Functions" Trigonometric Functions PPT Courseware (Basic Relationships of Trigonometric Functions with Same Angles in Lesson 2)

Part One: Learning Objectives

1. Understand and master the derivation and application of basic relational expressions of congruent trigonometric functions. (emphasis)

2. Be able to use basic relational expressions of congruent trigonometric functions to simplify, evaluate and prove identities. (difficulty)

core competencies

1. Carry out operations through the basic relations of congruent trigonometric functions and cultivate mathematical operation literacy.

2. Develop logical reasoning skills with the help of proofs of mathematical formulas.

PPT on the concept of trigonometric functions, part 2: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. square relationship

(1)Formula: sin2α+cos2α=_____.

(2) Verbal description: The sum of the squares of the sine and cosine of the same angle α is equal to _____.

2. quotient relationship

(1) Formula: sin αcos α=_____(α≠kπ+π2, k∈Z).

(2) Language description: The quotient of sine and cosine of the same angle α is equal to __________.

Thinking: For any angle α, is sin22α+cos22α=1 true?

Tip: Established. The emphasis on the same angle in the square relationship is arbitrary and has nothing to do with the expression of the angle.

First try

1. The result of simplifying 1-sin23π5 is ()

A． cos3π5　B． sin3π5

C． -cos3π5 D． -sin3π5

2. If α is an angle in the second quadrant, which of the following formulas is true ()

A． tan α＝-sin αcos α

B. cos α＝－1－sin2 α

C． sin α=-1-cos2 α

D. tan α＝cos αsin α

3. If cos α=35, and α is the fourth quadrant angle, then tan α=________.

The concept of trigonometric functions PPT, the third part of the content: cooperative exploration to improve literacy

Direct application of same-angle trigonometric functions to evaluate

[Example 1] (1) It is known that α∈π, 3π2, tan α=2, then cos α=________.

(2) It is known that cos α=-817, find the values ​​of sin α and tan α.

[Idea Tips] (1) Find cos α according to tan α=2 and sin2α+cos2α=1 series of equations.

(2) First determine which quadrant angle angle α is based on the known conditions, and then classify and discuss to find sin α and tan α.

regular method

Methods to solve the problem of evaluating values ​​by using the basic relations of congruent trigonometric functions:

1. Given a certain trigonometric function value of angle α, to find the remaining trigonometric function values ​​of angle α, attention should be paid to the reasonable selection of formulas. Generally, the square relationship is used first, and then the quotient relationship is used.

2. If the quadrant where the angle α is located has been determined, there is only one set of results when calculating the values ​​of the other two trigonometric functions; if the quadrant where the angle α is located is uncertain, it should be classified and discussed. Generally, there are two sets of results.

Reminder: When applying the square relationship to find the value of a trigonometric function, pay attention to the judgment of the position of the terminal side of the angle and determine the sign of the evaluated value.

Track training

1. It is known that sin α+3cos α=0, find the values ​​of sin α and cos α.

[Solution]　∵sin α＋3cos α＝0，

∴sin α＝-3cos α.

And sin2α+cos2α=1,

∴(-3cos α)2+cos2α=1,

That is, 10cos2α=1,

∴cos α＝±1010.

And from sin α=-3cos α,

It can be seen that sin α and cos α have different signs,

∴The terminal side of angle α is in the second or fourth quadrant.

When the terminal side of angle α is in the second quadrant, cos α=-1010, sin α=31010;

When the terminal side of angle α is in the fourth quadrant, cos α=1010, sin α=-31010.

Flexible application of same-angle trigonometric functions to evaluate relational expressions

[Example 2] (1) It is known that sin α+cos α=713, α∈(0, π), then tan α=________.

(2) It is known that sin α+cos α sin α-cos α=2, calculate the values ​​of the following formulas.

①3sin α-cos α2sin α+3cos α;

②sin2α-2sin αcos α+1.

[Ideas](1) Method 1: Find sin αcos α→Find sin α-cos α→Find sin α and cos α→Find tan α

Method 2: Find sin αcos α → String cut to construct an equation about tan α → Find tan α

(2) Find tan α→evaluate by substitution or string transformation

regular method

1. Among the three formulas sin α＋cos α, sin α-cos α, and sin αcos α, if one of them is known, the other two can be found, that is, "know one and find two". The relationship between them is: (sin α±cos α)2＝1±2sin αcos α.

2. Given that tan α=m, find the values ​​of the homogeneous expressions about sin α and cos α.

To solve this kind of problem, you need to pay attention to the following two points: (1) It must be a trigonometric function expression about the homogeneous formula of sin α and cos α (or can be converted into a homogeneous formula); (2) Because cos α≠0, it can Divide by cos α, so that the required expression can be transformed into an expression about tan α, and then substitute the value of tan α=m to complete the evaluation of the required expression.

Reminder: To find the value of sin α+cos α or sin α-cos α, pay attention to the position of the terminal side of the angle and use the trigonometric function line to determine their sign.

PPT on the concept of trigonometric functions, part 4: Achieve standards in class and solidify double bases

1. Thinking and analysis

(1) For any angle α, sin α2cos α2=tan α2 is true. ()

(2) Because sin2 94π+cos2 π4=1, sin2α+cos2β=1 is established, where α and β are arbitrary angles. ()

(3) For any angle α, sin α=cos α·tan α is true. ()

[ Tips

2. It is known that tan α=-12, then the value of 2sin αcos αsin2α-cos2α is ()

A.43B. 3

C． －43　D． -3

3. It is known that α is the second quadrant angle, tan α=-12, then cos α=________.

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Update Time: 2024-07-03

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