cos 15° = (√6 + √2) / 4
Answer (1)
To show that cos 1 5 ∘ = 4 6 + 2 , we can utilize the cosine angle subtraction formula. The formula is:
cos ( a − b ) = cos a cos b + sin a sin b
For 1 5 ∘ , we can use a = 4 5 ∘ and b = 3 0 ∘ . Therefore:
cos 1 5 ∘ = cos ( 4 5 ∘ − 3 0 ∘ ) = cos 4 5 ∘ cos 3 0 ∘ + sin 4 5 ∘ sin 3 0 ∘
Now, let's use known trigonometric values:
cos 4 5 ∘ = 2 2
cos 3 0 ∘ = 2 3
sin 4 5 ∘ = 2 2
sin 3 0 ∘ = 2 1
Substituting these values into the formula:
cos 1 5 ∘ = ( 2 2 ) ( 2 3 ) + ( 2 2 ) ( 2 1 )
Now, calculate each term:
2 2 × 2 3 = 4 6
2 2 × 2 1 = 4 2
Add these terms together:
cos 1 5 ∘ = 4 6 + 4 2 = 4 6 + 2
Hence, cos 1 5 ∘ = 4 6 + 2 is indeed correct.
