"Starting from the inclination of the ladder" The relationship between the sides and angles of a right triangle PPT courseware 5

Description

"Starting from the inclination of the ladder" The relationship between the sides and angles of a right triangle PPT courseware 5

Targeted

In a right triangle, if you know one side and an acute angle, can you find the other sides and angles?

Guess how tall this ancient tower is?

Think about it, can you use the mathematics knowledge you have learned to measure the height of this ancient tower?

think about it

Xiao Ming looked up at the top of the tower at point A and measured the size of ∠1. He then moved 50m toward the tower to point B and measured the size of ∠2. Based on these, he calculated the height of the tower. You know how he did it? Made?

Let’s start with the inclination of the ladder

Ladders are common objects in our daily lives

Can you compare the two ladders which one is steeper? What are your options?

sign of progress

From sensibility to rationality

The relationship between the sides and angles of a right triangle: the trigonometric function of acute angles - the tangent function

In a right triangle, if the ratio of the opposite side of an acute angle to the adjacent side is a constant value, then the value of the angle is also determined.

In Rt△ABC, the ratio of the opposite side to the adjacent side of acute angle A is called the tangent of ∠A, denoted as tanA, that is

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

Use mathematics to explain life

As shown in the figure, tangent is often used to describe the slope of a hillside. For example, if a hillside rises by 60m for every 100m it advances in the horizontal direction, then the slope i (i.e. tanα) of the hillside is:

i=tanα=60/100=3/5

Teacher Tips:

The angle (α) between the slope surface and the horizontal surface is called the slope angle, and the ratio of the vertical height and the horizontal width of the slope surface is called the slope i (or slope ratio), that is, the slope is equal to the tangent of the slope angle.

Practice in class

1. As shown in the figure, △ABC is an isosceles right triangle. Can you find tanC based on the data given in the figure?

2. As shown in the figure, a person walks 200m from point A at the foot of the mountain to point B at the top of the mountain. It is known that the vertical distance from the top of the mountain B to the foot of the mountain is 55m. Find the slope of the hillside (the result is accurate to 0.001m).

4. As shown in the figure, in Rt△ABC, the opposite side and adjacent side of the acute angle A are expanded 100 times at the same time, and the value of tanA ( )

A. Expand 100 times B. Reduce 100 times

C. unchanged D. uncertain

5. It is known that ∠A and ∠B are acute angles

(1) If ∠A=∠B, then tanA tanB;

(2) If tanA=tanB, then ∠A ∠B.

Several issues that should be noted in the definition:

1.tanA is defined in a right triangle, and ∠A is an acute angle (pay attention to the combination of numbers and shapes to construct a right triangle).

2.tanA is a complete symbol, representing the tangent of ∠A. It is customary to omit the "∠" sign;

3.tanA is a ratio (the ratio of the sides of a right angle. Pay attention to the order of the ratios, and tanA�0 has no unit.

4.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.

5. If the angles are equal, the tangent values ​​are equal; the tangent values ​​of two acute angles are equal, then the two acute angles are equal.

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Update Time: 2024-07-11

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