Exact Value of cos 36°

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


How to find exact value of cos 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, cos 36° = \(\frac{\sqrt{5} + 1}{4}\)

 Submultiple Angles






11 and 12 Grade Math

From Exact Value of cos 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. Fraction as a Part of Collection | Pictures of Fraction | Fractional

    Feb 24, 24 04:33 PM

    Pictures of Fraction
    How to find fraction as a part of collection? Let there be 14 rectangles forming a box or rectangle. Thus, it can be said that there is a collection of 14 rectangles, 2 rectangles in each row. If it i…

    Read More

  2. Fraction of a Whole Numbers | Fractional Number |Examples with Picture

    Feb 24, 24 04:11 PM

    A Collection of Apples
    Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One…

    Read More

  3. Identification of the Parts of a Fraction | Fractional Numbers | Parts

    Feb 24, 24 04:10 PM

    Fractional Parts
    We will discuss here about the identification of the parts of a fraction. We know fraction means part of something. Fraction tells us, into how many parts a whole has been

    Read More

  4. Numerator and Denominator of a Fraction | Numerator of the Fraction

    Feb 24, 24 04:09 PM

    What are the numerator and denominator of a fraction? We have already learnt that a fraction is written with two numbers arranged one over the other and separated by a line.

    Read More

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

    Feb 24, 24 10:59 AM

    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