find the 5 trig ratios when sin theta = -5/13 and lies in quadrant 3

make a sketch in III

y = -5, r = 13 , then
x^2 + (-5)^2 = 169
x = -12 in III

sinØ = -5/13, cscØ = -13/5
cosØ = -12/13, secØ = -13/12
tanØ = -5/-12 = 5/12
cotØ = 12/5

To find the trigonometric ratios of an angle, we need to know its reference angle and the quadrant in which it lies. In this case, we are given that the angle lies in quadrant 3 and its sine is -5/13.

In quadrant 3, the sine, cosine, and tangent are negative, while the cosecant, secant, and cotangent are positive.

To find the reference angle, we use the fact that the sine of an angle is equal to the sine of its reference angle. Since sin(theta) = -5/13, we have sin(ref angle) = -5/13.

To find the reference angle, we can use the inverse sine function (also known as arcsin) on a calculator. The inverse sine of -5/13 is approximately -0.3948 radians or -22.62 degrees.

Now, we can use the reference angle to find the trigonometric ratios in quadrant 3:

1. Sine (sin(theta)) = -5/13 (given)
2. Cosine (cos(theta)) = -cos(ref angle) = -cos(-0.3948) (use the negative sign in quadrant 3)
3. Tangent (tan(theta)) = -sin(ref angle) / cos(ref angle) = -(-5/13) / -cos(-0.3948)
4. Cosecant (csc(theta)) = 1 / sin(theta) = 1 / (-5/13)
5. Secant (sec(theta)) = 1 / cos(theta) = 1 / (-cos(-0.3948))
6. Cotangent (cot(theta)) = 1 / tan(theta) = 1 / [-(-5/13) / -cos(-0.3948)]

Calculating these values using a calculator will give you the specific numerical values for each trigonometric ratio.