Question

Find the value of cos 15°​

Answer 1

Answer:

or all values of the angle A we know that, (sin A2 + cos A2)2  = sin2 A2  + cos2 A2  + 2 sin A2 cos A2   = 1 + sin A  

Therefore, sin A2  + cos A2  = ± √(1 + sin A), [taking square root on both the sides]  

Now, let A = 30° then, A2 = 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 A2 - cos A2)2  = sin2 A2 + cos2 A2 - 2 sin A2 cos A2  = 1 - sin A  

Therefore, sin A2  - cos A2  = ± √(1 - sin A), [taking square root on both the sides]

Now, let A = 30° then, A2 = 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 (1√2 sin 15˚ - 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 ∙ 12

or, sin 15° - cos 15° = - √22

Thus, sin 15° - cos 15° < 0

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

Now, subtracting (iv) from (iii) we get,

2 cos 15° = 1+12−−−−−√+1−12−−−−−√

2 cos 15° = 3√+12√

cos 15° = 3√+122√

Therefore, cos 15° = 3√+122√

Answer 2

$\large \underline{ \underline{ \bold{ \: Answer : \: \: \: }}}$

$\to$ $\frac{\sqrt{3} + 1}{2 \sqrt{2} }$

$\large \underline{ \underline{ \bold{ \: Formula : \: \: \: }}}$

$\to \cos(a - b) = \cos(a) \cos(b) + \sin(a) \sin(b)$

$\large \underline{ \underline{ \bold{ \: Explaination : \: \: \: }}}$

$\to \cos(15) \\ \\ \to \cos(45 - 30) \\ \\ \to\cos(45) \cos(30) + \sin(45) \sin(30) \\ \\ \to\frac{1}{ \sqrt{2} } \times \frac{ \sqrt{3} }{2} + \frac{1}{ \sqrt{2} } \times \frac{ 1 }{ 2} \\ \\ \to \frac{ \sqrt{3} }{2 \sqrt{2} } + \frac{1}{2 \sqrt{2} } \\ \\ \to \frac{\sqrt{3} + 1}{2 \sqrt{2} }$