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




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.



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.

Share this page: What’s this?

Recent Articles

  1. Worksheet on Triangle | Homework on Triangle | Different types|Answers

    Jun 21, 24 02:19 AM

    Find the Number of Triangles
    In the worksheet on triangle we will solve 12 different types of questions. 1. Take three non - collinear points L, M, N. Join LM, MN and NL. What figure do you get? Name: (a)The side opposite to ∠L…

    Read More

  2. Worksheet on Circle |Homework on Circle |Questions on Circle |Problems

    Jun 21, 24 01:59 AM

    Circle
    In worksheet on circle we will solve 10 different types of question in circle. 1. The following figure shows a circle with centre O and some line segments drawn in it. Classify the line segments as ra…

    Read More

  3. Circle Math | Parts of a Circle | Terms Related to the Circle | Symbol

    Jun 21, 24 01:30 AM

    Circle using a Compass
    In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We…

    Read More

  4. Circle | Interior and Exterior of a Circle | Radius|Problems on Circle

    Jun 21, 24 01:00 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More

  5. Quadrilateral Worksheet |Different Types of Questions in Quadrilateral

    Jun 19, 24 09:49 AM

    In math practice test on quadrilateral worksheet we will practice different types of questions in quadrilateral. Students can practice the questions of quadrilateral worksheet before the examinations

    Read More