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प्रश्न
Find the value of the following expression:
sin 45° cos 30° – cos 45° sin 30°
योग
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उत्तर
Given: sin 45° cos 30° – cos 45° sin 30°
Step-wise calculation:
1. Use the sine difference identity:
sin (A – B) = sin A cos B – cos A sin B
So, the expression equals sin (45° – 30°) = sin 15°.
2. Compute sin 15° by substituting standard values:
`sin 45^circ = sqrt(2)/2`
`cos 30^circ = sqrt(3)/2`
`cos 45^circ = sqrt(2)/2`
`sin 30^circ = 1/2`
So, sin 45° cos 30° – cos 45° sin 30°
= `(sqrt(2)/2) (sqrt(3)/2) - (sqrt(2)/2) (1/2)`
= `(sqrt(2)/2) xx ((sqrt(3) - 1)/2)`
= `(sqrt(2) (sqrt(3) - 1))/4`
= `((sqrt(6) - sqrt(2)))/4`
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