Find the value of the function sin 14pi/10

I suppose you mean to find the value of the expression (not function) sin(14π/10).

To start, 14/10 = 7/5.

Next, expand sin(7x) using the angle sum identity,

This lets us write

and we can keep applying the identity to the multiple-angle argument to wind up with

We can replace to write this in terms of sin(x) only.

Then we can find the value of sin(7π/5) by plugging in π/5. But to do that, we need to first find the value of sin(π/5).

Using the same process as above, we have

When x = π/5, the left side reduces to sin(π) = 0. Let y = sin(π/5). Then

Factorize the right side:

Solving for y tells us that either y = 0 (which can't be true, because 0 < π/5 < π/2 means 0 < sin(π/5) < 1), or

Using the quadratic formula, we find

or 4 possible values, approximately -0.5878, 0.5878, -0.9512, or 0.9512.

Of course, only one of these values can be correct. We know sin(π/5) should be positive, so we throw out the two negative options. Also, sin(x) is strictly increasing over (0, π/2). This means 0 < π/5 < π/4 tells us 0 < sin(π/5) < sin(π/4), and sin(π/4) is approximately 0.707. Then sin(π/5) must be 0.5878, or exactly

Now plug this into the identity we found for sin(7x).