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° = $$\frac{\sqrt{3} - 1}{2\sqrt{2}}$$

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.

Recent Articles

1. Method of H.C.F. |Highest Common Factor|Factorization &Division Method

Apr 13, 24 05:12 PM

We will discuss here about the method of h.c.f. (highest common factor). The highest common factor or HCF of two or more numbers is the greatest number which divides exactly the given numbers. Let us…

2. Factors | Understand the Factors of the Product | Concept of Factors

Apr 13, 24 03:29 PM

Factors of a number are discussed here so that students can understand the factors of the product. What are factors? (i) If a dividend, when divided by a divisor, is divided completely

3. Methods of Prime Factorization | Division Method | Factor Tree Method

Apr 13, 24 01:27 PM

In prime factorization, we factorise the numbers into prime numbers, called prime factors. There are two methods of prime factorization: 1. Division Method 2. Factor Tree Method

4. Divisibility Rules | Divisibility Test|Divisibility Rules From 2 to 18

Apr 13, 24 12:41 PM

To find out factors of larger numbers quickly, we perform divisibility test. There are certain rules to check divisibility of numbers. Divisibility tests of a given number by any of the number 2, 3, 4…