How would I find the other 5 trigonometric functions?

Question

How would I be able to find the other 5 trigonometric functions of a problem like this?

cosΘ= -4/5, π < Θ < 3π /2

Answers are appreciated but solving steps would help so much more!! Thank you so much

Answer 1

Draw a QIII angle in standard position, and label the acute angle its terminal ray makes with the negative x-axis θ_ref (its reference angle). Draw a perpendicular from the terminal ray back to the negative x-axis. Since cosθ = -4/5, label the leg adjacent to θ_ref -4 and label the hypotenuse 5 (the hypotenuse (or radius) will ALWAYS be positive in these questions. By pythagorean theorem, the leg opposite θ_ref has a magnitude of 3, and because we are below the x-axis, we will label the leg -3. Now just use SOHCAHTOA trig, treating θ_ref as θ:

sinθ = - 3/5 , cscθ = - 5/3

cosθ = - 4/5 , secθ = - 5/4

tanθ = 3/4 , cotθ = 4/3

Notice how the signs (+ or -) agree with the mnemonic ASTC (only tangent is + in QIII).