Exact Value of cos 15°

How to find the exact value of cos 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, subtracting (iv) from (iii) we get,

2 cos 15° = $\sqrt{1 + \frac{1}{2}} + \sqrt{1 - \frac{1}{2}}$

2 cos 15° = $\frac{\sqrt{3} + 1}{\sqrt{2}}$

cos 15° = $\frac{\sqrt{3} + 1}{2\sqrt{2}}$

Therefore, cos 15° = $\frac{\sqrt{3} + 1}{2\sqrt{2}}$

● Submultiple Angles

• Trigonometric Ratios of Angle A2A2
• Trigonometric Ratios of Angle A3A3
• Trigonometric Ratios of Angle A2A2 in Terms of cos A
• tan A2A2 in Terms of tan A
• Exact value of sin 7½°
• Exact value of cos 7½°
• Exact value of tan 7½°
• Exact Value of cot 7½°
• Exact Value of tan 11¼°
  
• Exact Value of sin 15°
• Exact Value of cos 15°
• Exact Value of tan 15°
  
• Exact Value of sin 18°
• Exact Value of cos 18°
• Exact Value of sin 22½°
• Exact Value of cos 22½°
• Exact Value of tan 22½°
• Exact Value of sin 27°
• Exact Value of cos 27°
• Exact Value of tan 27°
  
• Exact Value of sin 36°
• Exact Value of cos 36°
• Exact Value of sin 54°
• Exact Value of cos 54°
• Exact Value of tan 54°
• Exact Value of sin 72°
• Exact Value of cos 72°
• Exact Value of tan 72°
• Exact Value of tan 142½°
  
• Submultiple Angle Formulae
  
• Problems on Submultiple Angles

11 and 12 Grade Math

From Exact Value of cos 15° to HOME PAGE