Solving Trig Functions: How Do You Solve For Sin 7.5 Using Half-Angle and Double-Angle Identities?

Answer 1

Formulas to remember:

sin(2a)=2sin(a)cos(a)
cos(2a)=cos²(a)-sin²(a)
From the second equation, replace a by a/2 and 2a by a to get
cos(a)=cos²(a/2)-sin²(a/2)
=2cos²(a/2)-1 ...(1)
=1-2sin²(a/2) ...(2)

From (1) we get
cos²(a/2)=(1/2)(1+cos(a))
From (2) we get
sin²(a/2)=(1/2)(1-cos(a))

So
sin(7.5)
=√((1/2)(1-cos(15))
=√((1/2)(1-√((1/2)(1+cos(30))))
=√((1/2)(1-√((1/2)(1+√(3)/2)))
=0.13...

Answer 2

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