The point lies on the terminal arm of an angle in standard position. What are the correct values of the following three ratios?

Question

BlackTom AI · Accepted Answer

We’re given the point P(-6, -8) on the terminal arm of an angle in standard position, so the reference radius r is the distance from the origin to P.

First, compute r: r = sqrt((-6)^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10.

From there, the basic trigonometric values are determined:
- cos(θ) = x/r = -6/10 = -3/5.
- sin(θ) = y/r = -8/10 = -4/5, so csc(θ) = r/y = 1/sin(θ) = -5/4.
- tan(θ) = y/x = (-8)/(-6) = 4/3 since both coordinates are negative, placing θ in the third quadrant where both sine and cosine are negative and tangent is positive.

Now evaluate each option:
- Option a: cos(θ) = -4/5, csc(θ) = 5/8, tan(θ) = 4/6.
  This fails because cos(θ) should be -3/5, not -4/5. Also csc(θ) should be -5/4 (not +5/8), and tan(θ) should be 4/3 (not 4/6). The signs and fractions do not align with the calculated values.
- Option b: cos(θ) = -3/5, csc(θ) = -5/4, tan(θ) = 4/3.
  This matches all computed results exactly: cos is -3/5, csc is -5/4, and tan is 4/3. The signs are consistent with the point in the third quadrant, where x and y are negative and tan is positive.
- Option c: cos(θ) = -6/10, csc(θ) = 5/4, tan(θ) = 6/8.
  Here cos is presented as -6/10 which simplifies to -3/5 (that part would be okay), but csc is +5/4 while the correct csc is -5/4, and tan should be 4/3, not 6/8. This option mixes incorrect signs and incorrect tangent value.
- Option d: cos(θ) = 3/5, csc(θ) = -8/5, tan(θ) = -8/6.
  This has cos with the wrong sign (positive instead of negative), csc with incorrect magnitude and sign, and tan with the wrong ratio. These deviations contradict the actual coordinates and radius.

In summary, the only set that aligns with the computed trigonometric values derived from P(-6, -8) and the radius 10, while respecting quadrant II/III signs (cos negative, sin negative, tan positive), is the combination shown in option b.

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