The point P(-4, -5) is on the terminal arm of an angle .

Question

Calculate the exact values for the six trigonometric ratios.

Answer 1

Josh,

 

If you plot the point P in the xy-plane, you see it lies in Quadrant III. Draw the segment from the origin O to P and find its length:

 

r = √((-4)²+(-5)²) = √(16 + 25) = √41 .

 

You already know that x = -4 and y = -5. You can now find the six values for the trigonometric ratios:

 

sin θ = y/r = -5/√41 = -(5√41)/41

cos θ = x/r = -4/√41 = -(4√41)/41

tan θ = y/x = -5/-4 = 5/4

cot θ = x/y = -4/-5 = 4/5

sec θ = r/x = √41/(-4) = -(√41)/4

csc θ = r/y = √41/(-5) = -(√41)/5

 

Remember that θ (spelled "theta") is the angle between the positive x-axis and the terminal side (ray OP). I rationalized the denominators because that is usually required when answering questions like this.

 

Let me know if you have any questions about this.