"Starting from the inclination of the ladder" PPT courseware 2 on the relationship between the sides and angles of a right triangle

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"Starting from the inclination of the ladder" PPT courseware 2 on the relationship between the sides and angles of a right triangle

Tangent The relationship between the sides and angles of a right triangle: Trigonometric functions of acute angles - 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 side opposite to the adjacent side of acute angle A

It is called the tangent of ∠A, denoted as tanA, that is, tanA=opposite side of ∠A/adjacent side of ∠A

think about it

As shown in the figure, when an acute angle A in Rt△ABC is determined, the ratio of its opposite side to the adjacent side is determined. At this time, are the ratios between other sides also determined?

in conclusion:

In Rt△ABC, if the acute angle A is determined, then the ratio of the opposite side to the hypotenuse and the ratio of the adjacent side to the hypotenuse of ∠A are also determined.

Sine and Cosine

In Rt△ABC, the ratio of the opposite side to the hypotenuse of acute angle A is called the sine of ∠A, denoted as sinA, that is, sinA=opposite side of ∠A/hypotenuse of ∠A

In Rt△ABC, the ratio of the adjacent side and the hypotenuse of acute angle A is called the cosine of ∠A, which is recorded as cosA, that is, cosA=the adjacent side of ∠A/the hypotenuse of ∠A

The sine, cosine, and tangent of acute angle A are all trigonometric functions of ∠A.

True knowledge is born in practice

1. As shown in the figure: In isosceles △ABC, AB=AC=5, BC=6.

Find: sinB,cosB,tanB.

Teacher’s tip: Pass through point A and draw AD perpendicular to BC and D.

2. In Rt△ABC, ∠C=900, BC=20, find: the perimeter of △ABC.

3. 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 sinA ( )

A. Expand 100 times B. Reduce 100 times C. No change D. Uncertain

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

(1) If ∠A=∠B, then sinA____sinB;

(2) If sinA=sinB, then ∠A____∠B.

Several issues that should be noted in the definition:

1. sinA, cosA, tanA are 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. sinA, cosA, and tanA are each a complete symbol, representing the sine, cosine and tangent of ∠A respectively. It is customary to omit "∠" in the notation;

3. sinA, cosA, and tanA are each a ratio. Pay attention to the order of the ratios, and sinA, cosA, and tanA are all greater than 0 and have no unit.

4. The sizes of sinA, cosA, and tanA are only related to the size of ∠A and have nothing to do with the side length of the right triangle.

5. If the angles are equal, then the values ​​of their trigonometric functions are equal; if the values ​​of the trigonometric functions of two acute angles are equal, then the two acute angles are equal.

sublimation of knowledge

1. As shown in the figure, find the sine, cosine and tangent of ∠α and ∠β respectively.

2. In △ABC, AB=5, BC=13, AD is the height on the side of BC, AD=4. Find: CD, sinC.

3. In Rt△ABC, ∠BCA=90°, CD is the center line, BC=8, CD=5.

Find sin∠ACD, cos∠ACD and tan∠ACD.

4. In Rt△ABC, ∠C=90°, what is the relationship between sinA and cosB?

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Update Time: 2024-07-28

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