Evaluate the expression under the given conditions (show steps please)

sin(è/2); tan è = − 5 / 12 , è in Quadrant IV

5^2+12^2 = h^2

25+144 = h^2 = 169
so hypotenuse = h = 13
sin e = -5/13
cos e = 12/13

sin(e/2) = -sqrt( [1-cos e] /2 )
sin(e/2) = -sqrt (1/13/2) = -sqrt(1/26)

Thank you!!!!

To evaluate the expression sin(θ/2) with the given conditions, we first need to find the value of θ/2.

Given that tan(θ) = -5/12 and θ is in Quadrant IV, we can determine the sine of θ/2 as follows:

1. Start by finding the value of θ. Since tan(θ) = -5/12 and θ is in Quadrant IV, we know that the tangent is negative in this quadrant. We can use the inverse tangent function to find θ:

θ = arctan(-5/12)

2. Use a calculator to find the value of arctan(-5/12). This will give you the measure of θ.

θ ≈ -21.80°

3. Now that we have the value of θ, we can find θ/2:

θ/2 = -21.80° / 2

θ/2 ≈ -10.90°

4. Finally, we can evaluate the expression sin(θ/2):

sin(θ/2) = sin(-10.90°)

5. Use a calculator or reference table to find the sine of -10.90°:

sin(-10.90°) ≈ -0.187 (rounded to three decimal places)

Therefore, the value of sin(θ/2) under the given conditions is approximately -0.187.