"Starting from the inclination of the ladder" The relationship between the sides and angles of a right triangle PPT courseware 3

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"Starting from the inclination of the ladder" The relationship between the sides and angles of a right triangle PPT courseware 3

Teacher's message

The acute trigonometric function describes the relationship between the sides and angles of a right triangle. It is also an important functional relationship between variables. It is both novel and charming. You should establish a good relationship with it!

Explore new knowledge in practice

As the ladder ascends and becomes steeper, what happens to the inclination angle and the ratio of the vertical height to the horizontal width?

Let’s start with the inclination of the ladder

What should Xiao Ming do if he cannot successfully measure the distance B1 C1 from the top of the ladder to the base of the wall due to his height, and thus cannot describe the inclination of the ladder? Do you have any tips?

think about it

(1) What is the relationship between right triangle AB1C1 and right triangle AB2C2?

(2) What is the relationship between B1C1/AC1 and B2C2/AC2?

(3) What if the position of B2 on the ladder is changed? What conclusion can you draw from this?

∠tangent of A

In Rt△ABC, if the acute angle A is determined, then the ratio of the opposite side of ∠A to the adjacent side is determined. This ratio is called the tangent of ∠A.

Noted as: tanA

tanA=opposite side of ∠A/adjacent side of ∠A

Thinking: We discussed the inclination of the ladder earlier. Is there any relationship between the inclination of the ladder and tanA?

Discuss

As shown in the figure, is the inclination of ladder AB1 related to tanA? Is it related to ∠A?

Related to tanA: The larger the value of tanA, the steeper the ladder AB1.

Related to ∠A: The larger ∠A is, the steeper the ladder AB1 is.

A few notes on the definition:

1) In junior high school, tangent is defined in a right triangle, and ∠A is an acute angle.

2) tanA is a complete symbol, which represents the tangent of ∠A. It is customary to omit the angle symbol "∠" in the notation. But the tangent of ∠BAC is expressed as: tan∠BAC. The tangent of ∠1 is expressed as: tan∠1.

3) tanA�0 has no unit. It represents a ratio, that is, the ratio of the opposite side to the adjacent side of the acute angle ∠A in a right triangle (note the order: opposite/adjacent).

4) tanA does not mean "tan" times "A".

5) The size of tanA is only related to the size of ∠A and has nothing to do with the side length of the right triangle.

Review, reflect, and deepen:

1. The definition of tangent.

2. The relationship between the inclination of the ladder and tanA. (The relationship between ∠A and tanA).

3. The method of combining numbers and shapes; the awareness of constructing a right triangle.

4. “General → Special → General” mathematical thinking method.

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Update Time: 2024-07-11

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