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.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

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

Shiv Puran In Odia Pdf Link -

After extensive research, we have identified the most reliable sources for the Shiv Puran in Odia PDF link. Please note that these are external links to public resources:

The Shiv Puran in Odia PDF is a gateway to understanding Lord Shiva’s infinite compassion and power. Whether you are a student of scriptures, a devout Shaivite, or a curious reader, this digital copy allows you to carry the divine wisdom of Kailash wherever you go.

Download your copy today and let the stories of Bholenath inspire your spiritual journey.

Har Har Mahadev!

If you found this post helpful, please share it with other Odia Shiva devotees. Bookmark this page for updated links.

Searching for a "solid story" from the Shiva Purana translated into Odia, the most accessible and authentic digital version is the Sankhipta Shiva Purana published by Gita Press Gorakhpur. 📖 Shiva Purana in Odia: Digital Access

You can read and download the complete text via these primary digital archives: Internet Archive - Sankhipta Shiva Purana (Odia)

: This is a high-quality scan of the Gita Press edition, available for free download in PDF format. Scribd - Bruhat Siba Purana shiv puran in odia pdf link

: A comprehensive version translated by D. Das and K.M. Pradhan, available for online reading. Internet Archive - Shiva Purana Katha Saar

: A condensed summary of the main stories extracted from the larger Mahapurana. ✨ A "Solid Story" from the Shiva Purana

One of the most powerful and popular stories found in these texts is the origin of the Shiva Linga (The Pillar of Fire):

The Story: Long ago, Lord Brahma (the Creator) and Lord Vishnu (the Preserver) began to argue over who was superior. Their dispute grew so intense that it threatened the stability of the universe. Suddenly, a massive, endless pillar of fire (Jyotirlinga) appeared between them.

To find its end, Brahma took the form of a swan and flew upward, while Vishnu became a boar and dug deep into the earth. Neither could find the limit. Vishnu returned and humbly admitted his failure. Brahma, however, lied and claimed he found the top, using a Ketaki flower as a false witness. Shiva then emerged from the pillar, exposing Brahma's lie and declaring that He is the supreme source from which all creation begins and ends. 🕉️ Related Odia Spiritual Resources

If you are looking for specific prayers or shorter texts in Odia, you can also find:

Shiva Panchakshara Stotra: A prayer focused on the "Na-Ma-Si-Va-Ya" syllables. After extensive research, we have identified the most

Shiva Manasa Puja: A mental worship ritual dedicated to Lord Shiva. ?

Finding a digital version of a sacred text like the Shiv Puran (Shiva Purana) in Odia can be a powerful way to connect with your spiritual roots on the go. You can find free digital copies of the Odia Shiv Puran on platforms like Internet Archive.

Below is a review of the Odia version of this sacred scripture to help you decide if it’s the right addition to your collection. Review: Shiv Puran (Odia Version)

The Shiv Puran is one of the 18 Mahapuranas of Hinduism, primarily focusing on the glory, philosophy, and legends of Lord Shiva and Goddess Parvati. The Odia translation brings these ancient Sanskrit teachings to life for Odia-speaking devotees, often maintaining the traditional poetic flow common in regional religious literature. 1. Content & Depth

The text is typically divided into several Samhitas (sections), such as the Vidyeshwara Samhita (focused on Shiva worship and the Panchakshari mantra) and the Rudra Samhita (covering the creation of the universe and Shiva’s divine acts). For many readers, the Odia edition—particularly those from Gita Press, Gorakhpur—is praised for its clarity and preservation of the original "bhakti" (devotional) essence. 2. Accessibility & Language

The Pros: The translation is designed to be accessible for personal study or communal reading during religious ceremonies. It provides a comprehensive spiritual guide for those interested in Shaiva-Advaita philosophy.

The Cons: Some readers find certain "large" or "Bruhat" editions a bit challenging if they are written in a strictly classical poetic format, which can be harder to digest for those accustomed to modern prose. 3. Key Benefits for Readers Devotees often read the Shiv Puran for: Linga Purana in Odia - Ritikart If you found this post helpful, please share

The Shiv Puran (or Shiva Purana) is one of the eighteen Mahapuranas of Hinduism, primarily dedicated to Lord Shiva and Goddess Parvati. For Odia-speaking devotees, the Shiv Puran in Odia serves as a vital spiritual guide, translating ancient Sanskrit wisdom into a language that is deeply integrated into Odisha's rich Shaivite culture. Direct Download Links for Shiv Puran in Odia PDF

You can find digital versions of the Shiv Puran in Odia through the following reputable digital archives:

Sankhipta Shiv Puran (Gita Press): A concise version published by the world-renowned Gita Press, Gorakhpur. Download from Internet Archive

Bruhat Siba Purana (Vol 1): A detailed edition translated by D. Das and K.M. Pradhan. Download from Internet Archive Read on Scribd Importance and Contents of the Shiv Puran

The Shiv Puran was originally written in Sanskrit by Romaharshana, a disciple of Maharishi Ved Vyasa. The text traditionally consists of 100,000 verses divided into twelve Samhitas (sections), though many modern versions are abridged for easier reading. Key Sections (Samhitas):

The National Digital Library of India (ndl.iitkgp.ac.in) sometimes hosts regional religious texts. Search for "Odia Shiv Purana" with filters for "Religious Studies."

In Odia tradition, listening to the Shiv Puran (Shiv Purana Patha) is considered as holy as reading it. If you cannot find a perfect PDF, consider:

When you download a religious text, ensure it is authentic:

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.