use the unit circle to find the exact value for all the trig functions of è=2ð/3.

I do not understand your notation, è=2ð/3.

Do you mean theta = 2 pi/3?

The sine of 2 pi/3 is the same as the sine of pi/3, which is (sqrt3)/2

The cosine of 2 pi/3 has the same absolute value as cos pi/3 (1/2), but has opposite sign

To find the exact values for all the trigonometric functions of the angle θ = 2π/3 using the unit circle, follow these steps:

Step 1: Draw the unit circle
Start by drawing a circle with a radius of 1 unit. This circle represents the unit circle.

Step 2: Mark the reference angle
For the angle θ = 2π/3, the reference angle is π/3. Mark this angle on the unit circle.

Step 3: Determine the coordinates
Since the radius of the unit circle is 1, the coordinates of the point on the unit circle corresponding to the reference angle π/3 are (cos(π/3), sin(π/3)). The x-coordinate is cos(π/3) and the y-coordinate is sin(π/3).

Step 4: Evaluate the trigonometric functions
Using the coordinates from Step 3, we can determine the values of the trigonometric functions:

- Sine (sin(θ)): sin(2π/3) = sin(π/3) = y-coordinate = √3/2
- Cosine (cos(θ)): cos(2π/3) = cos(π/3) = x-coordinate = 1/2
- Tangent (tan(θ)): tan(2π/3) = tan(π/3) = sin(θ)/cos(θ) = (√3/2) / (1/2) = √3

Note that the other three trigonometric functions (cosecant, secant, and cotangent) can be determined by taking the reciprocals of the sine, cosine, and tangent, respectively.

- Cosecant (csc(θ)): csc(2π/3) = csc(π/3) = 1 / sin(θ) = 1 / (√3/2) = 2/√3
- Secant (sec(θ)): sec(2π/3) = sec(π/3) = 1 / cos(θ) = 1 / (1/2) = 2
- Cotangent (cot(θ)): cot(2π/3) = cot(π/3) = 1 / tan(θ) = 1 / √3

Therefore, the exact values for all the trigonometric functions of θ = 2π/3 are:

sin(2π/3) = √3/2
cos(2π/3) = 1/2
tan(2π/3) = √3
csc(2π/3) = 2/√3
sec(2π/3) = 2
cot(2π/3) = 1/√3