Exact Value of sin 36°

We will learn to find the exact value of sin 36 degrees using the formula of multiple angles.


How to find exact value of sin 36°?

Let A = 18°                          

Therefore, 5A = 90° 

⇒ 2A + 3A = 90˚

⇒ 2θ = 90˚ - 3A

Taking sine on both sides, we get 

sin 2A = sin (90˚ - 3A) = cos 3A 

⇒ 2 sin A cos A = 4 cos\(^{3}\) A - 3 cos A

⇒ 2 sin A cos A - 4 cos\(^{3}\) A + 3 cos A = 0 

⇒ cos A (2 sin A - 4 cos\(^{2}\) A + 3) = 0 

Dividing both sides by cos A = cos 18˚ ≠ 0, we get

⇒ 2 sin θ - 4 (1 - sin\(^{2}\) A) + 3 = 0

⇒ 4 sin\(^{2}\) A + 2 sin A - 1 = 0, which is a quadratic in sin A

Therefore, sin θ = \(\frac{-2 \pm \sqrt{- 4 (4)(-1)}}{2(4)}\)

⇒ sin θ = \(\frac{-2 \pm \sqrt{4 + 16}}{8}\)

⇒ sin θ = \(\frac{-2 \pm 2 \sqrt{5}}{8}\)

⇒ sin θ = \(\frac{-1 \pm \sqrt{5}}{4}\)

Now sin 18° is positive, as 18° lies in first quadrant.

Therefore, sin 18° = sin A = \(\frac{-1 \pm \sqrt{5}}{4}\)

Now, cos 36° = cos 2 ∙ 18°

⇒ cos 36° = 1 - 2 sin\(^{2}\) 18°

⇒ cos 36° = 1 - 2\((\frac{\sqrt{5} - 1}{4})^{2}\)

⇒ cos 36° = \(\frac{16 - 2(5 + 1 - 2\sqrt{5})}{16}\)

⇒ cos 36° = \(\frac{1 + 4\sqrt{5}}{16}\)

⇒ cos 36° = \(\frac{\sqrt{5} + 1}{4}\)

Therefore, sin 36° = \(\sqrt{1 - cos^{2} 36°}\),[Taking sin 36° is positive, as 36° lies in first quadrant, sin 36° > 0]

⇒ sin 36° = \(\sqrt{1 - (\frac{\sqrt{5} + 1}{4})^{2}}\)

⇒ sin 36° = \(\sqrt{\frac{16 - (5 + 1 + 2\sqrt{5})}{16}}\)

⇒ sin 36° = \(\sqrt{\frac{10 - 2\sqrt{5}}{16}}\)

⇒ sin 36° = \(\frac{\sqrt{10 - 2\sqrt{5}}}{4}\)

Therefore, sin 36° = \(\frac{\sqrt{10 - 2\sqrt{5}}}{4}\)

 Submultiple Angles






11 and 12 Grade Math

From Exact Value of sin 36° to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. 4th Grade Mental Math on Roman Numerals | Roman Numerals Quiz

    Feb 23, 24 03:55 PM

    In 4th grade mental math on numbers, students can practice different questions on write the Hindu-Arabic numerals, write the Roman Numerals, comparison of roman numerals, addition of roman numerals.

    Read More

  2. 4th Grade Mental Math on Numbers | Mental Math 4th Grade with Answers

    Feb 23, 24 02:24 PM

    4th Grade Mental Math on Numbers
    In 4th grade mental math on numbers, students can practice different questions on numbers in figures, number name, place value, face value, comparison of number and formation of greatest and smallest…

    Read More

  3. Roman Numerals | System of Numbers | Symbol of Roman Numerals |Numbers

    Feb 23, 24 01:28 PM

    List of Roman Numerals Chart
    How to read and write roman numerals? Hundreds of year ago, the Romans had a system of numbers which had only seven symbols. Each symbol had a different value and there was no symbol for 0. The symbol…

    Read More

  4. Worksheet on Roman Numerals |Roman Numerals|Symbols for Roman Numerals

    Feb 22, 24 04:15 PM

    Roman Numbers Table
    Practice the worksheet on roman numerals or numbers. This sheet will encourage the students to practice about the symbols for roman numerals and their values. Write the number for the following: (a) V…

    Read More

  5. Roman Symbols | What are Roman Numbers? | Roman Numeration System

    Feb 22, 24 02:30 PM

    Roman Numbers
    Do we know from where Roman symbols came? In Rome, people wanted to use their own symbols to express various numbers. These symbols, used by Romans, are known as Roman symbols, Romans used only seven…

    Read More