To find the sine, cosine, and tangent of the angle 8π/6, we can start by simplifying the angle. The fraction 8π/6 simplifies to 4π/3, because both the numerator and the denominator can be divided by 2.
Next, we can recognize that 4π/3 radians is in the third quadrant of the unit circle. In this quadrant, the sine function is negative, while the cosine function is also negative. Let’s find the reference angle for 4π/3:
Reference angle = 4π/3 – π = 4π/3 – 3π/3 = π/3.
Now, we can find the sine and cosine for the reference angle π/3, which are:
- sin(π/3) = √3/2
- cos(π/3) = 1/2
Since 4π/3 is in the third quadrant, we apply the signs for sine and cosine in this quadrant:
- sin(4π/3) = -√3/2
- cos(4π/3) = -1/2
Finally, we can calculate the tangent, which is defined as the ratio of sine to cosine:
- tan(4π/3) = sin(4π/3) / cos(4π/3) = (-√3/2) / (-1/2) = √3.
In summary, the sine, cosine, and tangent of 8π/6 (or 4π/3) are:
- Sine: -√3/2
- Cosine: -1/2
- Tangent: √3