8
Chay Den Ben Em Voi Van Toc 493km Vietsub Instant
THERE ARE TWO special triangles in trigonometry. One is the 30°-60°-90° triangle. The other is the isosceles right triangle. They are special because with simple geometry we can know the ratios of their sides, and therefore solve any such triangle.
Theorem. In a 30°-60°-90° triangle the sides are in the ratio
1 : 2 : 
.
We will prove that below.
Note that the smallest side, 1, is opposite the smallest angle, 30°; while the largest side, 2, is opposite the largest angle, 90°. (Theorem 6). (For, 2 is larger than 
. Also, while 1 : : 2 correctly corresponds to the sides opposite 30°-60°-90°, many find the sequence 1 : 2 : easier to remember.)
The cited theorems are from the Appendix, Some theorems of plane geometry.
Here are examples of how we take advantage of knowing those ratios. First, we can evaluate the functions of 60° and 30°.
Example 1. Evaluate cos 60°.
Answer. For any problem involving a 30°-60°-90° triangle, the student should not use a table. The student should sketch the triangle and place the ratio numbers.
Since the cosine is the ratio of the adjacent side to the hypotenuse, we can see that
cos 60° = ½.
Example 2. Evaluate sin 30°.
Answer. According to the property of cofunctions, sin 30° is equal to cos 60°. sin 30° = ½.
On the other hand, you can see that directly in the figure above.
Problem 1. Evaluate sin 60° and tan 60°.
To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
The sine is the ratio of the opposite side to the hypotenuse.
| sin 60° = |  2 |
= ½ . |
The tangent is ratio of the opposite side to the adjacent.
| tan 60° = | 1 |
=  . |
Problem 2. Evaluate cot 30° and cos 30°.
The cotangent is the ratio of the adjacent side to the opposite.
| Therefore, on inspecting the figure above, cot 30° = | 1 |
= .
Or, more simply, cot 30° = tan 60°.
As for the cosine, it is the ratio of the adjacent side to the hypotenuse. Therefore,
| cos 30° = | 2 |
= ½. |
Before we come to the next Example, here is how we relate the sides and angles of a triangle:
If an angle is labeled capital A, then the side opposite will be labeled small a. Similarly for angle B and side b, angle C and side c.
Example 3. Solve the right triangle ABC if angle A is 60°, and side AB is 10 cm.
Solution. To solve a triangle means to know all three sides and all three angles. Since this is a right triangle and angle A is 60°, then the remaining angle B is its complement, 30°.
Again, in every 30°-60°-90° triangle, the sides are in the ratio 1 : 2 : , as shown on the left.
When we know the ratios of the sides, then to solve a triangle we do not require the trigonometric functions or the Pythagorean theorem. We can solve it by the method of similar figures.
Now, the sides that make the equal angles are in the same ratio. Proportionally,
2 : 1 = 10 : AC.
2 is two times 1. Therefore 10 is two times AC. AC is 5 cm.
The side adjacent to 60°, we see, is always half the hypotenuse.
As for BC—proportionally,
2 : = 10 : BC.
To produce 10, 2 has been multiplied by 5. Therefore, will also be multiplied by 5. BC is 5 cm.
In other words, since one side of the standard triangle has been multiplied by 5, then every side will be multiplied by 5.
1 : 2 : = 5 : 10 : 5.
Compare Example 11 here.
Again: When we know the ratio numbers, then to solve the triangle the student should use this method of similar figures, not the trigonometric functions.
(In Topic 10, we will solve right triangles whose ratios of sides we do not know.)
Problem 3. In the right triangle DFE, angle D is 30° and side DF is 3 inches. How long are sides d and f ?
The student should draw a similar triangle in the same orientation. Then see that the side corresponding to was multiplied by .
Therefore, each side will be multiplied by . Side d will be
1
= . Side f will be 2.
Problem 4. In the right triangle PQR, angle P is 30°, and side r is 1 cm. How long are sides p and q ?
The side corresponding to 2 has been divided by 2. Therefore, each side must be divided by 2. Side p will be ½, and side q will be ½.
Problem 5. Solve the right triangle ABC if angle A is 60°, and the hypotenuse is 18.6 cm.
The side adjacent to 60° is always half of the hypotenuse -- therefore, side b is 9.3 cm.
But this is the side that corresponds to 1. And it has been multiplied by 9.3. Therefore, side a will be multiplied by 9.3.
It will be 9.3 cm.
Problem 6. Prove: The area A of an equilateral triangle whose side is s, is
A = ¼
s2.
The area A of any triangle is equal to one-half the sine of any angle times the product of the two sides that make the angle. (Topic 2, Problem 6.)
In an equilateral triangle each side is s , and each angle is 60°. Therefore,
A = ½ sin 60°s2.
Since sin 60° = ½,
A = ½· ½ s2 = ¼s2.
Problem 7. Prove: The area A of an equilateral triangle inscribed in a circle of radius r, is
| A = | 3 4 |
|
r2. |
Chay Den Ben Em Voi Van Toc 493km Vietsub Instant
The specific speed of mentioned in the title is not a random figure; it represents the world record for the fastest badminton smash. This record was set by Malaysian player Tan Boon Heong in 2013 during a controlled test of equipment. In the context of the drama, it serves as a metaphor for the intensity and "terminal velocity" of the passion and love between the two main characters. Plot and Themes
Trái ngược với Tae-yang, Tae-joon coi cầu lông chỉ là một công việc kiếm sống thông thường. Sinh ra trong gia đình có truyền thống kinh doanh thiết bị thể thao, anh chàng tham gia đấu giải cốt chỉ để kiếm tiền thưởng chứ không hề có tham vọng lớn lao. Tuy nhiên, cuộc gặp gỡ định mệnh với Tae-yang tại câu lạc bộ mới đã đánh thức ngọn lửa đam mê ngủ quên trong anh, thúc đẩy anh muốn trở thành chỗ dựa vững chắc cho cô cả trên sân đấu lẫn ngoài đời thực.
Minh is a street racer turned delivery driver, trying to leave his past behind. One night, he’s called to pick up a package from an old address—his ex-girlfriend, Lan, who now works as a night-shift pharmacist.
“Or the speed I used to chase you,” Minh replies.
The evolving relationship between Tae-joon and Tae-yang is a slow-burn romance filled with tension, humor, and heartfelt moments. Their partnership begins with mutual frustration but gradually transforms into deep trust and affection, making every milestone in their relationship feel earned and meaningful. Chay den Ben Em Voi Van Toc 493km Vietsub
If you have ever searched for Vietnamese subtitled Korean dramas, you have likely encountered the phrase "Chạy đến bên em với vận tốc 493km Vietsub". This keyword is the Vietnamese title for the 2022 KBS2 romantic sports drama , which is also known by its international English title "Love All Play" . Combining the intensity of a sports competition with the sweetness of a romantic comedy, this drama quickly gained a dedicated following among Vietnamese and international viewers. But what exactly is this drama about, and why should you add it to your watchlist?
Tình cảm giữa hai nhân vật chính bùng nổ nhanh và mạnh mẽ như một cú đập cầu uy lực trên sân đấu. Tóm Tắt Nội Dung Phim
Tiêu đề phim gây ấn tượng mạnh bởi con số cụ thể: 493 km/h. Trong bộ môn cầu lông, đây là được thiết lập vào năm 2013 bởi vận động viên người Malaysia Tan Boon Heong.
Having gained recognition in "Extracurricular" and "Mouse," Park Ju-hyun delivers a powerful performance as a former prodigy fighting to reclaim her place in the world of badminton. The specific speed of mentioned in the title
Hiện nay, khán giả Việt Nam có thể dễ dàng tìm kiếm và thưởng thức bộ phim với chất lượng cao (Full HD, 4K) cùng bản dịch tiếng Việt (Vietsub) chuẩn xác thông qua các nền tảng bản quyền trực tuyến sau:
LET ALL PLAY - CHẠY ĐẾN BÊN EM VỚI VẬN TỐC 493KM
Một vận động viên vốn dĩ chỉ xem cầu lông là một công việc kiếm sống bình thường do gia đình anh kinh doanh thiết bị cầu lông. Sau khi bị cắt khỏi đội cũ và gia nhập đội Eunice, anh gặp Tae-yang. Chính mong muốn gây ấn tượng với cô đã khơi dậy niềm đam mê thể thao thực thụ trong anh. Diễn Biến & Mối Quan Hệ
Bộ phim xoay quanh câu chuyện tình lãng mạn, hài hước và đầy đam mê giữa hai nhân vật chính: và Park Tae Yang . Họ là những vận động viên cầu lông thuộc đội tuyển kinh doanh, những người sống và thở cùng đam mê cầu lông. Plot and Themes Trái ngược với Tae-yang, Tae-joon
"Chay den Ben Em Voi Van Toc 493km Vietsub" is more than just a sports drama. It is a story about the speed of life, the weight of expectations, and the powerful force of human connection. Whether you are drawn to the electrifying badminton matches, the palpable chemistry between the leads, or the deeply human story of getting back up after a fall, this drama has something for everyone.
Phim gửi gắm thông điệp sâu sắc: Ai cũng có quyền mắc sai lầm, nhưng điều quan trọng là cách chúng ta đứng dậy . Hành trình Park Tae-yang đối mặt với lỗi lầm quá khứ, vượt qua sự tẩy chay của đồng đội để tìm lại hào quang là nguồn cảm hứng lớn cho người xem.
Bạn đã xem những tập đầu của bộ phim này chưa, hay bạn đang cần uy tín để xem bản HD không quảng cáo?
Problem 8. Prove: The angle bisectors of an equilateral triangle meet at a point that is two thirds of the distance from the vertex of the triangle to the base.
Let ABC be an equilateral triangle, let AD, BF, CE be the angle bisectors of angles A, B, C respectively; then those angle bisectors meet at the point P such that AP is two thirds of AD.
First, triangles BPD, APE are congruent.
For, since the triangle is equilateral and BF, AD are the angle bisectors, then angles PBD, PAE are equal and each
30°;
and the side BD is equal to the side AE, because in an equilateral triangle the angle bisector is the perpendicular bisector of the base.
Angles PDB, AEP then are right angles and equal.
Therefore,
triangles BPD, APE are congruent.
| Now, | BP PD |
= csc 30° = 2. |
Therefore, BP = 2PD.
But AP = BP, because triangles APE, BPD are conguent, and those are the sides opposite the equal angles.
Therefore, AP = 2PD.
Therefore AP is two thirds of the whole AD.
Which is what we wanted to prove.
The proof
Here is the proof that in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : . It is based on the fact that a 30°-60°-90° triangle is half of an equilateral triangle.
Draw the equilateral triangle ABC. Then each of its equal angles is 60°. (Theorems 3 and 9)
Draw the straight line AD bisecting the angle at A into two 30° angles.
Then AD is the perpendicular bisector of BC (Theorem 2). Triangle ABD therefore is a 30°-60°-90° triangle.
Now, since BD is equal to DC, then BD is half of BC.
This implies that BD is also half of AB, because AB is equal to BC. That is,
BD : AB = 1 : 2
From the Pythagorean theorem, we can find the third side AD:
| AD2 + 12 | = | 22 |
| AD2 | = | 4 − 1 = 3 |
| AD | = | . |
Therefore in a 30°-60°-90° triangle the sides are in the ratio 1 : 2 : ; which is what we set out to prove.
Corollary. The square drawn on the height of an equalateral triangle is three fourths of the square drawn on the side.
Next Topic: The Isosceles Right Triangle
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