"Arbitrary Angle and Radians System" Trigonometric Functions PPT Courseware (Lesson 2: Radians System)

Description

"Arbitrary Angle and Radians System" Trigonometric Functions PPT Courseware (Lesson 2: Radians System)

Part One: Learning Objectives

1. Understand the one-to-one correspondence between the set of angles and the set of real numbers in the radian system.

2. Understand the definition of "angle in radians", master the conversion of radians and angles, arc length formula and sector area formula, and be familiar with the number of radians of special angles. (main difficulty)

3. Understand the difference and connection between "angle system" and "radian system". (easy to make mistakes)

core competencies

1. Cultivate students’ mathematical abstract literacy through learning the concept of radian system.

2. Improve students' mathematical operation literacy with the help of conversion between radian system and angle system.

PPT on arbitrary angles and radians, part 2: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. Two systems of units for measuring angles

(1) Angle system:

①Definition: The system of units that uses ____ as the unit to measure angles.

②An angle of 1 degree: ____ of the circumferential angle.

(2) Radian system:

①Definition: The unit system that measures angles with ____ as the unit.

②An angle of 1 radian: the central angle subtended by an arc whose length is equal to ____.

2. Calculation of radians

Thinking: Is the ratio lr related to the radius of the circle?

Tip: The ratio of arc length to radius corresponding to a certain central angle α is uniquely determined and has nothing to do with the radius.

3. Conversion between degrees and radians

4. Correspondence between some special angles and radians

5. Arc length and area formulas of sectors

Suppose the radius of the sector is R, the arc length is l, and α (0＜α＜2π) is its central angle, then

(1)Arc length formula: l=____.

(2) Sector area formula: S=____=____.

First try

1. Which of the following conversion results is wrong ()

A． 60° converted into radians is π3 rad

B． -103π rad degree of transformation is -600°

C. -150° converted into radians is -76π rad

D.π12 rad degree of transformation is 15°

2.29π6 is ()

A． first quadrant angle

B． second quadrant angle

C. third quadrant angle

D. Fourth quadrant angle

3. (1)7π5 rad converted into angle is ________.

(2)The number of radians of 105° is ________.

4. The area of ​​a sector with radius 2 and central angle π6 is _________.

Any angle and radian system PPT, the third part of the content: cooperative exploration to improve literacy

Interaction and application of angles and radians

[Example 1] (1) ① Convert 112°30′ into radians to ________.

②Convert -5π12rad into an angle of ________.

(2) It is known that α=15°, β=π10 rad, γ=1 rad, θ=105°, φ=7π12 rad. Try to compare the sizes of α, β, γ, θ and φ.

regular method

The key and method of converting the angle system and the radian system into each other

1 Key: Grasping the mutual transformation formula π rad = 180° is the key;

2 method: Degrees × π180 = radians; radians × 180π° = degrees;

3. When converting angles into radians, you should first convert minutes and seconds into degrees and then into radians.

Express angle in radians

[Example 2] (1) The set of angle α where the terminal side passes through the point (a, a) (a≠0) is ()

A.π4

B.π4, 5π4

C.αα＝π4＋2kπ，k∈Z

D.αα=π4+kπ, k∈Z

(2) Use radians to express the set of angles θ where the terminal edge falls within the shaded part (excluding the boundary) as shown in the figure.

regular method

1. Representation of an angle in radians that has the same terminal side as angle α:

In the radian system, the angle with the same terminal side as angle α can be expressed as {β|β=2kπ+α, k∈Z}, that is, the angle with the same terminal side as angle α can be expressed as α plus an integer multiple of 2π ．

2. Steps to write the set of area angles based on a known figure:

(1) Observe the graphics carefully.

(2) Write the boundary of the region as the representation of the hour angle of the terminal edge.

(3) Use inequalities to express angles within the area.

Reminder: The angle system and the radian system cannot be mixed.

Class summary

1. When expressing angles, due to the advantages of the radian system, radian is often used to express angles, but it should also be noted that when using radian to express angles, it cannot be mixed with the angle system.

2. When applying the arc length and sector area formulas in the radian system, please note that the prerequisite for use is the radian system. At the same time, attention should also be paid to the combination with other knowledge such as function content.

PPT for arbitrary angles and radians, part 4: reaching the standard and solidifying the double base in court

1. Thinking and Analysis()

(1) An angle of 1 radian is 1360 of the circumferential angle.

(2) An angle of 1 radian is greater than an angle of 1 degree.

2. The radius of the circle is r. The central angle subtended by the arc with length 32r on the circle is ()

A.23rad

B.32rad

C.2π3 rad

D.3π2 rad

3. If -570° is written in the form of 2kπ＋α(k∈Z,0≤α＜2π), then α=________.

4. Find the arc length and area of ​​a sector with radius π cm and central angle 120°.

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Update Time: 2024-10-05

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

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