"Arbitrary Angle and Radians System" Trigonometric Functions PPT (Second Lesson System of Radians)

Description

"Arbitrary Angle and Radians System" Trigonometric Functions PPT (Second Lesson System of Radians)

Part One: Learning Objectives

Understand the concept of radians

Ability to convert between angles and radians

Able to express angles with the same terminal sides in radians

Understand the arc length and area formulas of sectors in the radian system

PPT on arbitrary angles and radians, part 2: independent learning

Problem guide

Preview the textbook P172-P175 and think about the following questions:

1.1 How is an angle in radians defined?

2. How to convert radians to angles?

3. What are the formulas for the arc length and area of ​​a sector in radians?

A preliminary exploration of new knowledge

1. Two systems of measuring angles

■Instructions from famous teachers

(1) When expressing the size of an angle in radians, you can omit "radians" or "rad" and just write the number of radians corresponding to the angle. For example, the angle α = -3.5 rad can be written as α = -3.5 .When expressing the size of an angle in angle units, "degree" or "°" cannot be omitted.

(2) Whether the size of the angle is measured in radians or degrees, it is a fixed value that has nothing to do with the size of the radius.

2. Calculation and conversion of radians

(1) Calculation of radians

(2) Interchange between radian and angle

3. The arc length and area formula of the sector in the radian system (r is the radius of the circle where the sector is located, n is the central angle of the sector)

■Instructions from famous teachers

(1) When applying the sector area formula S=12|α|r2, it should be noted that the unit of α is "radians".

(2) From any two quantities among α, r, l and S, the other two quantities can be obtained.

self-test

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

(1)The angle of 1 rad is larger than the angle of 1°. ()

(2) Measuring angles using the angle system and the radian system are both related to the radius of the circle. ()

(3) Each angle in the radian system has a unique angle corresponding to it in the angle system. ()

(4) An angle of 1° is 1360 of the circumferential angle, and an angle of 1 rad is 12π of the circumferential angle. ()

8π5 radians converted into angle is ()

A． 278°B. 280°

C. 288° D. 318°

The area of ​​a sector with radius 2 and central angle π3 is ()

A.4π3 B. π

C.2π3 D.π3

Any angle and radian system PPT, the third part of the content: interactive teaching and practice

Interchange between angle system and radian system

Convert the following angles to radians:

(1)37°30′; (2)-216°; (3)7π12; (4)-11π5.

regular method

The principle of mutual transformation between angle system and radian system

(1) Principle: Keep in mind 180°=π rad, and make full use of 1°=π180 rad and 1 rad=180π° for conversion.

(2) Method: Suppose the number of radians of an angle is α and the number of angles is n, then α rad = α·180π°; n° = n·π180rad.

Track training

1. Convert the following angles into radians.

(1)-1 500°=________.

(2)67°30′=________.

2. Convert the following radians into angles.

(1)23π6=________.

(2)-13π6=________.

Express angles with the same terminal sides in radians

Write -1 480° in the form of 2kπ+α(k∈Z), where 0≤α<2π, and determine which quadrant angle it is?

regular method

Express two points of interest with the same terminal angle in radians.

(1) When the angle 2kπ + α (k∈Z) with the same terminal sides is expressed in radians, 2kπ is an even multiple of π, not an integer multiple.

(2) Note that the angle system and the radian system cannot be mixed.

PPT for arbitrary angles and radians, part 4: feedback on compliance with standards

1. The angle that is the same as the terminal side of 60° can be expressed as ()

A． k·360°＋π3(k∈Z)

B． 2kπ＋60°(k∈Z)

C. 2k·360°＋60°(k∈Z)

D. 2kπ+π3(k∈Z)

2.1 The angle of 920° converted into radians is ()

A.163B.323

C.163π D.323π

3. In a circle with a radius of 8 cm, the length of the arc subtended by the central angle of 5π3 is ()

A.403π cm B.203π cm

C.2003π cm D.4003π cm

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Update Time: 2024-07-03

This template belongs to Mathematics courseware People's Education High School Mathematics Edition A Compulsory Course 1 industry PPT template

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