Exact Value of sin 15°

How to find the exact value of sin 15° using the value of sin 30°?

Solution:

For all values of the angle A we know that, (sin \(\frac{A}{2}\) + cos \(\frac{A}{2}\))\(^{2}\)  = sin\(^{2}\) \(\frac{A}{2}\)  + cos\(^{2}\) \(\frac{A}{2}\)  + 2 sin \(\frac{A}{2}\) cos \(\frac{A}{2}\)   = 1 + sin A 

Therefore, sin \(\frac{A}{2}\)  + cos \(\frac{A}{2}\)  = ± √(1 + sin A), [taking square root on both the sides]

Now, let A = 30° then, \(\frac{A}{2}\) = \(\frac{30°}{2}\) = 15° and from the above equation we get,

sin 15° + cos 15° = ± √(1 + sin 30°)                       ….. (i)

Similarly, for all values of the angle A we know that, (sin \(\frac{A}{2}\) - cos \(\frac{A}{2}\))\(^{2}\)  = sin\(^{2}\) \(\frac{A}{2}\) + cos\(^{2}\) \(\frac{A}{2}\) - 2 sin \(\frac{A}{2}\) cos \(\frac{A}{2}\)  = 1 - sin A 

Therefore, sin \(\frac{A}{2}\)  - cos \(\frac{A}{2}\)  = ± √(1 - sin A), [taking square root on both the sides]

Now, let A = 30° then, \(\frac{A}{2}\) = \(\frac{30°}{2}\) = 15° and from the above equation we get,

sin 15° - cos 15°= ± √(1 - sin 30°)                  …… (ii)

Clearly, sin 15° > 0 and cos 15˚ > 0

Therefore, sin 15° + cos 15° > 0

Therefore, from (i) we get,

sin 15° + cos 15° = √(1 + sin 30°)                                  ..... (iii)

Again, sin 15° - cos 15° = √2 (\(\frac{1}{√2}\) sin 15˚ - \(\frac{1}{√2}\) cos 15˚)

or, sin 15° - cos 15° = √2 (cos 45° sin 15˚ - sin 45° cos 15°)

or, sin 15° - cos 15° = √2 sin (15˚ - 45˚)

or, sin 15° - cos 15° = √2 sin (- 30˚)

or, sin 15° - cos 15° = -√2 sin 30°

or, sin 15° - cos 15° = -√2 ∙ \(\frac{1}{2}\)

or, sin 15° - cos 15° = - \(\frac{√2}{2}\)

Thus, sin 15° - cos 15° < 0

Therefore, from (ii) we get, sin 15° - cos 15°= -√(1 - sin 30°)    ..... (iv)

Now, adding (iii) and (iv) we get,

2 sin 15° = \(\sqrt{1 + \frac{1}{2}} - \sqrt{1 - \frac{1}{2}}\)

2 sin 15° = \(\frac{\sqrt{3} - 1}{\sqrt{2}}\)

sin 15° = \(\frac{\sqrt{3} - 1}{2\sqrt{2}}\)

Therefore, sin 15° = \(\frac{\sqrt{3} - 1}{2\sqrt{2}}\)

● Submultiple Angles





11 and 12 Grade Math

From Exact Value of sin 15° 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

RSS

  1. Rupees and Paise | Paise Coins | Rupee Coins | Rupee Notes

    Dec 04, 23 02:14 PM

    Different types of Indian Coins
    Money consists of rupees and paise; we require money to purchase things. 100 paise make one rupee. List of paise and rupees in the shape of coins and notes:

    Read More

  2. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Dec 04, 23 01:50 PM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More

  3. The Story about Seasons | Spring | Summer | Autumn | Winter

    Dec 04, 23 01:49 PM

    The Four Seasons
    Kids let’s enjoy the story about seasons. Here we will discuss about the four seasons and the duration. Some months are too hot and some are too cold. The period of hot months is called the hot

    Read More