Exact Value of sin 72°

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

How to find the exact value of sin 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 = $\frac{-1 \pm \sqrt{5}}{4}$

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

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

And cos 18° = √(1 - sin$^{2}$ 18°), [Taking positive value, cos 18° > 0]

⇒ cos 18° = $\sqrt{1 - (\frac{\sqrt{5} - 1}{4})^{2}}$

⇒ cos 18° = $\sqrt{\frac{16 - (5 + 1 - 2\sqrt{5})}{16}}$

⇒ cos 18° = $\sqrt{\frac{10 + 2\sqrt{5}}{16}}$

Therefore, cos 18° = $\frac{\sqrt{10 + 2\sqrt{5}}}{4}$

Now sin 72° = sin (90° - 18°) = cos 18° = $\frac{\sqrt{10 + 2\sqrt{5}}}{4}$

11 and 12 Grade Math

From Exact Value of sin 72° 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.

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?

[?]Subscribe To This Site

E-mail Address

First Name

Then Don't worry — your e-mail address is totally secure. I promise to use it only to send you Math Only Math.