To find the sine, cosine, and tangent of the angle 8π/6, we first simplify the angle since it is greater than 2π. We can convert this angle into a more manageable form by using the periodicity of the trigonometric functions, which have a period of 2π:
1. **Simplify the angle:**
8π/6 can be simplified to 4π/3 (by dividing the numerator and denominator by 2). This new angle can be found on the unit circle.
2. **Determine the reference angle:**
The angle 4π/3 lies in the third quadrant. Its reference angle can be calculated as:
4π/3 – π = 4π/3 – 3π/3 = π/3
3. **Find sine and cosine values:**
In the third quadrant, the sine function is negative and the cosine function is also negative. The sine and cosine values associated with the reference angle π/3 are:
sin(π/3) = √3/2
cos(π/3) = 1/2
Therefore, in the third quadrant, they are:
sin(4π/3) = -√3/2
cos(4π/3) = -1/2
4. **Calculate tangent:**
The tangent function is defined as the ratio of sine to cosine. Thus:
tan(8π/6) = sin(4π/3) / cos(4π/3) = (-√3/2) / (-1/2) = √3
In summary, the values are:
sin(8π/6) = -√3/2
cos(8π/6) = -1/2
tan(8π/6) = √3