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Question

Find the value of sin75using sum or difference identity.


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Solution

Find the value of the given trigonometric ratio

Given trigonometric ratio: sin75

sin75can be expressed as, sin75=sin(45+30)

We know that sin(A+B)=sinAcosB+cosAsinB

So by applying the above sum identity we get,

sin75=sin45cos30+cos45sin30

As, sin45=12,cos30=32=,cos45=12 and sin30=12.

By substituting the values we get,

sin75=12×32+12×12=322+122=3+122

Hence, the value of sin75is 1+322.

Alternate solution:

sin75can be expressed as, sin75=sin(135-60)

We know that sin(A-B)=sinAcosB-cosAsinB

So by applying the above sum identity we get,

sin75=sin135cos60-cos135sin60

As, sin135=12,cos60=12,cos135=-12 and sin60=32.

By substituting the values we get,

sin75=12×12--12×32=122+322=1+322sin75=1+322


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