find:
sin 23pi/12
i got totally the wrong answer how do i approach this?
That is pi/12 less than 2 pi (360 degrees), which would be 345 or -15 degrees. The sin of that angle is -0.2588
If you don't want the decimal approximation, it is
-sin(45 - 30) = -sin 45 cos 30 + sin 30 cos 45 = (sqrt 2)/2*[1/2 - (sqrt3)/2]
= (sqrt2)/4 -(sqrt6)/4
haha i got that too! but the right answer is
(-root(2-root(3)))/2
why don't you evaluate your "right" answer and compare it with drwls answer ?
MMMhhh ??
To find the value of sin(23π/12), you can use the unit circle and trigonometric identities.
1. Start by converting the angle from degrees to radians. To convert from degrees to radians, use the formula: radians = degrees × (π/180). In this case, the angle is 23π/12 radians.
2. Recall that the unit circle represents the values of sine, cosine, and other trigonometric functions. Draw a unit circle and mark the angle of 23π/12 on it.
3. Note that 23π/12 radians is equivalent to 11π/12 + π/6 radians. This can be visualized by considering that 23π/12 = 12π/12 + 11π/12 = π + 11π/12 = π/6 + 11π/12.
4. From the unit circle, locate the corresponding angle of 11π/12 and π/6. Notice that 11π/12 is between π/6 and π/2. π/6 corresponds to the point (1/2, √3/2) on the unit circle, and π/2 corresponds to the point (0, 1).
5. Now, use the periodicity property of sine function. Since sinθ = sin(2π + θ) = sin(4π + θ) = ..., you can add or subtract full revolutions (2π, 4π, etc.) without changing the value of sine. In this case, adding 2π to 11π/12 will bring it in the range of π/6 to π/2.
6. Subtract 2π from 11π/12 to get -13π/12. This angle corresponds to the same point on the unit circle as 11π/12.
7. Since the value of sine in the second quadrant is positive, we can now find the value of sin(-13π/12) instead.
8. From the unit circle, locate the angle of -13π/12. It is equivalent to 11π/12 counterclockwise from the positive x-axis.
9. Use the symmetry property of the unit circle to find the coordinates for the angle -13π/12. The symmetric point to (1/2, √3/2) is (-1/2, √3/2). Therefore, sin(-13π/12) = √3/2.
So, sin(23π/12) is equal to sin(-13π/12), which is √3/2.
Note: When dealing with angles larger than 2π, it's essential to utilize the periodicity and symmetry properties of trigonometric functions to bring the angle into a range where the unit circle can be effectively used.