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प्रश्न
If sin (A − B) = sin A cos B − cos A sin B and cos (A − B) = cos A cos B + sin A sin B, find the values of sin 15° and cos 15°.
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उत्तर
Given:
sin (A − B) = sin A cos B − cos A sin B ......(1)
cos (A − B) = cos A cos B + sin A sin B ......(2)
`To find:
The values of `sin 15^@` and `cos 15^@`
In this problem, we need to find `sin 15^@` and `cos 15^@`
Hence to get `15^@` angle we need to choose the value if A and B such that `(A - B) = 15^@`
So If we choose A = 45° and B = 30°
Then we get (A - B) = 15°
Therefore by substituting A = 45° and B = 30° in equation (1)
We get
`sin(45^@ - 30^@) = sin 45^@ cos 30^@ - cos 45^@ sin 30^@`
Therefore
`sin(15^@) = sin 45^@ cos 30^@ - cos 45^@ sin 30^@` ....(3)
Now we know that,
`sin 45^@ = cos 45^@ = 1/sqrt2, sin 30^@ = 1/2, cos 30^@ = sqrt3/2`
Now by substituting above values in equation (3)
We get,
`sin (15^@) = (1/sqrt2) xx (sqrt3/2) - (1/sqrt2) xx (1/2)`
`= sqrt3/(2sqrt2) - 1/(2sqrt2)`
`= (sqrt3 - 1)/(2sqrt2)`
Therefore
`cos(15^@) = (sqrt3 -1)/(2sqrt2)` ....(6)
Therefore from equation (4) and (6)
`sin(15^@) = (sqrt3 - 1)/(2sqrt2)`
`cos(15^@) = (sqrt3 + 1)/(2sqrt2)`
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