I don't see how to do this without a calculator
sin ( - pi/12)
csc ( (25 pi)/12)
I'm obviously suppose to get - sin 2 pi
ok - sin ( (24 pi)/12)
I don't see how to get that though
To calculate trigonometric functions without using a calculator, it is helpful to remember some key values and identities. Here's how you can find the values of the given trigonometric functions:
1. sin(-π/12):
The first step is to recognize that sin(-x) = -sin(x). So, sin(-π/12) = -sin(π/12).
To find the value of sin(π/12), we can use the half-angle formula for sin:
sin(π/12) = √[(1 - cos(π/6)) / 2]
Now, let's find cos(π/6):
We know that cos(x) = sin(π/2 - x), so cos(π/6) = sin(π/2 - π/6) = sin(π/3) = √3/2.
Using this value, we get:
sin(π/12) = -√[(1 - (√3/2)) / 2]
= -√[(2 - √3) / 4]
Hence, sin(-π/12) = -sin(π/12) = -(-√[(2 - √3) / 4]) = √[(2 - √3) / 4].
2. csc((25π)/12):
The cosecant (csc) function is the reciprocal of the sine function: csc(x) = 1/sin(x).
To find csc((25π)/12), we need to find sin((25π)/12). We can use the sum-formula for sine to simplify this expression.
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
For (25π)/12, we can split it as a = π and b = π/12:
sin((25π)/12) = sin(π + (π/12))
= sin(π)cos(π/12) + cos(π)sin(π/12)
= 0*cos(π/12) + (-1)*sin(π/12)
= -sin(π/12)
Therefore, csc((25π)/12) = 1/sin((25π)/12) = 1/(-sin(π/12)).
We already found sin(π/12) in the previous calculation, so we can substitute that value here:
csc((25π)/12) = 1/[√[(2 - √3) / 4]].
These are the exact values for sin(-π/12) and csc((25π)/12) without using a calculator.