use appropriate compound angle formula to determine exact value of 19pi/12

Answer 1

To find the exact value of 19π/12, we can use the compound angle formula for sine.

The compound angle formula for sine is:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

In this case, A = 3π/4 and B = π/6.

Plugging these values into the formula, we get:
sin(19π/12) = sin(3π/4 + π/6)
= sin(9π/12 + 2π/12)
= sin(11π/12)

Now, we need to express the angle 11π/12 in terms of angles that we know the exact value of.

11π/12 can be expressed as 12π/12 - π/12, which is equivalent to π - π/12.

Now, we can use the compound angle formula for sine again to find the exact value of sin(11π/12).

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

In this case, A = π and B = π/12.

Plugging these values into the formula, we get:
sin(11π/12) = sin(π - π/12)
= sin(π)cos(π/12) - cos(π)sin(π/12)
= 0 - (-1)(√2/4)
= √2/4

So, the exact value of 19π/12 is √2/4.