Find the value of sin 75 using sum or difference identity

We have to find the value of sin 75 using sum or difference identity

Solution

Sin 75 we can write it as

Sin 75 = Sin(45+30)…………………..(1)

By applying the formula

Sin (A + B) = Sin A. Cos B + Cos A. Sin B

Sin (45 + 30) = Sin 45. Cos 30 + Cos 45. Sin 30…………………..(2)

Sin Values

sin 0° = √(0/4) = 0

sin 30° = √(1/4) = ½

sin 45° = √(2/4) = 1/√2

sin 60° = √3/4 = √3/2

cos 90° = √(4/4) = 1

Cos Values

cos 0° = √(4/4) = 1

cos 30° = √(3/4) = √3/2

cos 45° = √(2/4) = 1/√2

cos 60° = √(1/4) = 1/2

cos 90° = √(0/4) = 0

Substitute the value of sin 30, sin 45, cos 30 and cos 45 degrees, then equation (2) will becomes

Sin (45 + 30) = Sin 45. Cos 30 + Cos 45. Sin 30

Sin (45 + 30) = 1/√2 . √3/2 + 1/√2 . 1/2

Sin (45 + 30) = (√3 + 1) / 2√2

Hence the value of Sin 75 degree is equal to (√3 + 1) / 2√2.

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