Exact Value of cos 72°

We will learn to find the exact value of cos 72 degrees using the formula of submultiple angles.


How to find the exact value of cos 72°?

Let, A = 18°                        

Therefore, 5A = 90° 

⇒ 2A + 3A = 90˚

⇒ 2A = 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 A - 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 A = \(\frac{-2 \pm \sqrt{- 4 (4)(-1)}}{2(4)}\)

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

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

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

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

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

Now, cos 72° = cos (90° - 18°) = sin 18° = \(\frac{√5 - 1}{4}\)



 Submultiple Angles





11 and 12 Grade Math

From Exact Value of cos 72° 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. Multiplication of a Number by a 3-Digit Number |3-Digit Multiplication

    Mar 28, 24 11:44 AM

    Multiplying by 3-Digit Number
    In multiplication of a number by a 3-digit number are explained here step by step. Consider the following examples on multiplication of a number by a 3-digit number: 1. Find the product of 36 × 137

    Read More

  2. Multiply a Number by a 2-Digit Number | Multiplying 2-Digit by 2-Digit

    Mar 27, 24 05:21 PM

    Multiply 2-Digit Numbers by a 2-Digit Numbers
    How to multiply a number by a 2-digit number? We shall revise here to multiply 2-digit and 3-digit numbers by a 2-digit number (multiplier) as well as learn another procedure for the multiplication of…

    Read More

  3. Multiplication by 1-digit Number | Multiplying 1-Digit by 4-Digit

    Mar 26, 24 04:14 PM

    Multiplication by 1-digit Number
    How to Multiply by a 1-Digit Number We will learn how to multiply any number by a one-digit number. Multiply 2154 and 4. Solution: Step I: Arrange the numbers vertically. Step II: First multiply the d…

    Read More

  4. Multiplying 3-Digit Number by 1-Digit Number | Three-Digit Multiplicat

    Mar 25, 24 05:36 PM

    Multiplying 3-Digit Number by 1-Digit Number
    Here we will learn multiplying 3-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. 1. Multiply 201 by 3 Step I: Arrange the numb…

    Read More

  5. Multiplying 2-Digit Number by 1-Digit Number | Multiply Two-Digit Numb

    Mar 25, 24 04:18 PM

    Multiplying 2-Digit Number by 1-Digit Number
    Here we will learn multiplying 2-digit number by 1-digit number. In two different ways we will learn to multiply a two-digit number by a one-digit number. Examples of multiplying 2-digit number by

    Read More