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.

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.)

Here are examples of how we take advantage of knowing those ratios. First, we can evaluate the functions of 60° and 30°.

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.

Answer. According to the property of cofunctions, sin 30° is equal to cos 60°. sin 30° = ½.

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1 : 2: SqRt 3

The cotangent is the ratio of the adjacent side to the opposite.

Therefore, on inspecting the figure above, cot 30° =

= .

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° =

= 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,

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.

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 ?

1 : 2: SqRt 3

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 ?

Problem 5. Solve the right triangle ABC if angle A is 60°, and the hypotenuse is 18.6 cm.

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

Fun With Dick And Jane 720p Hindi Now

Nostalgia & Laughs: Exploring "Fun with Dick and Jane" (2005) in 720p Hindi

For fans of heist comedies and Jim Carrey’s signature physical humor, Fun with Dick and Jane remains a staple of mid-2000s cinema. As streaming habits evolve and global audiences expand, the search for specific versions of the film—such as the 720p Hindi dubbed version—has become increasingly popular among Indian viewers and the South Asian diaspora.

Final Verdict: Is It Worth Your Time?

Absolutely. Fun with Dick and Jane isn’t high art. It’s not trying to be The Big Short or Margin Call. It’s a 90-minute rapid-fire comedy about two nice people who become terrible criminals out of sheer desperation. Jim Carrey’s physical humor, ridiculously quotable lines, and a tight script make it a timeless watch. fun with dick and jane 720p hindi

Classic Comedy: The 1977 original stars Jane Fonda and George Segal as an affluent couple who turn to robbery to maintain their lifestyle. Nostalgia & Laughs: Exploring "Fun with Dick and

The Plot: From Suburbia to Stick-ups

Dick Harper (Jim Carrey) is a successful aerospace engineer living the American dream with his wife, Jane (Téa Leoni), and their son in a posh suburban neighborhood. When Dick is finally promoted to Vice President of Communications, he convinces Jane to quit her job to live a life of leisure. The Promotion Scene – Dick’s ecstatic monologue about

The Promotion Scene – Dick’s ecstatic monologue about finally making it, immediately followed by Globodyne’s stock crashing. The Hindi voice actor captures Carrey’s rapid shift from joy to panic flawlessly.
The "Learn to Rob" Montage – Dick and Jane practicing stick-ups in their living room. The Hindi translation of their awkward dialogue adds extra cringe-comedy.
The Spanish-Only Convenience Store – A gag about a clerk who only speaks Spanish becomes hilariously repurposed in Hindi with a Bhojpuri-speaking shopkeeper.
The Climactic Heist – Sneaking into the CEO’s mansion. The dubbed banter between Dick and Jane feels like a classic Bollywood husband-wife argument.

The Indian entertainment industry has experienced tremendous growth in recent years, driven by the increasing demand for digital content. The rise of streaming platforms, social media, and online YouTube channels has created new opportunities for creators to produce and distribute content that caters to diverse interests and preferences.

While the film was primarily released in English, it has been made available with various language options on streaming platforms over time. Streaming: You can find the movie on platforms like Amazon Prime Video Language Support:

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.

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.

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.

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.

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.