Evaluate giving exact values and identities. Calculator based solutions are not acceptable. Thanks for any help
sin (-pi/12)
Carmen, Bradley -- please use the same name when you post similar questions.
Also -- does your instructor accept answers from anonymous people on the internet?
yes
well pi is 180 degrees
180/12 = 15
so we really want sin(-15 degrees)
But I know that sin 30 = 1/2
and that cos 30 = (1/2) sqrt 3
and I know how to find the sine of half an angle if I have the cosine of the whole angle
sin(-15)º = sin(30-45)º
can you take it from there?
To evaluate sin(-π/12), we need to use the unit circle and the trigonometric identity.
1. Draw a unit circle with the positive x-axis as the reference line.
2. Find the angle corresponding to -π/12. Since -π/12 is negative, we go clockwise from the positive x-axis.
-π/12 lies between -π/6 (-30 degrees) and -π/4 (-45 degrees).
3. Determine the reference angle, which is the positive acute angle between the terminal side of the angle and the x-axis. In this case, the reference angle is π/12 (15 degrees).
4. Use the reference angle to find the value of sine. Since sine is positive in the second and third quadrants, we need to determine the value of sin(π/12).
sin(π/12) can be rewritten as sin(15 degrees).
5. Find the exact value of sin(15 degrees) by using the special right triangle or trigonometric identity.
One way to find the exact value is using the half-angle identity for sine, which states that sin(θ/2) = ±√[(1 - cos(θ)) / 2].
In this case, we have sin(15 degrees) = sin(30/2) = sin(30/2) = ±√[(1 - cos(30)) / 2].
The exact value of cos(30) can be found by using the special right triangle with sides 1, 2, and √3. Since cos(30) = √3/2, we can substitute this value in the equation above.
sin(15 degrees) = ±√[(1 - √3/2) / 2].
6. Simplify the expression.
To simplify, rationalize the denominator by multiplying both the numerator and denominator by √2.
sin(15 degrees) = ±√[(2 - √3) / 4].
So, the exact value of sin(-π/12) is ±√[(2 - √3) / 4].