Quadrants and Angles (page 2 of 3)

Sections: Introduction, Worked Examples (and Sign Chart), More Examples


  • Find the six trigonometric ratios for the angle having point (4, –3) on its terminal side.
     

     

    First, I'll draw a picture:

     

    x,y-axes with terminal side of angle theta in fourth quadrant; point (4, -3) on side

    My picture doesn't need to be exact or "to scale". I just need the general idea of what quadrant I'm in and where the angle θ ("theta") is.

     

    Now that I've drawn the angle in the fourth quadrant, I'll drop the perpendicular down from the axis:

     

    same picture, with perpendicular dropped from x-axis to point (4, -3), forming right triangle

     

     

    From the x- and y-values of the point they gave me (that I've drawn on the hypotenuse), I can label two of the sides of my right triangle:

     

    triangle labelled with base 4 and height -3

     

    Then the Pythagorean Theorem gives me the length r of the hypotenuse:

      r2 = 42 + (–3)2
      r2 = 16 + 9 = 25

      r = 5

    Now I'll finish my picture by adding the length of the hypotenuse to my right triangle:

     

     

    hypotenuse labelled with length 5

    And that's all I need for finding my ratios! To find my answers, I can just read the number from my picture:

      sin(theta) = -3/5, cos(theta) = 4/5, tan(theta) = -3/4, csc(theta) = -5/3, sec(theta) = 5/4, cot(theta) = -4/3

  • Determine the quadrant in which lies the terminal side of the angle theta,
    given that
    tan(θ) < 0 and sin(θ) < 0.
  • For this exercise, I need to consider the x- and y-values in the various quadrants, in the context of the trig ratios. I don't need to find any actual values; I only need to work with the signs and what I know about the ratios and the quadrants.

    The tangent ratio is y/x, so the tangent will be negative when x and y have opposite signs. This occurs in the second quadrant (where x is negative but y is positive) and in the fourth quadrant (where x is positive but y is negative). So the sign on the tangent tells me that the end of the angle is in QII or in QIV.

    The sine ratio is y/r, and the hypotenuse r is always positive. So the sine will be negative when y is negative, which happens in the third and fourth quadrants.

    So the tangent is negative in QII and QIV, and the sine is negative in QIII and QIV. The overlap between the two solutions is QIV, so:

      The terminal side of the angle θ lies somewhere in QIV.


The thought process for the exercise above leads to a rule for remembering the signs on the trig ratios in each of the quadrants. In the first quadrant, all the values (x, y, and r) are positive, so All the trig ratios are positive. In the second quadrant, the x-values are negative, so x/r and y/x are negative; only y/r is positive, so only the Sine is positive in QII. In the third quadrant, each of x and y is negative, so x/r and y/r are negative; only y/x is positive, so only the Tangent is positive in QIII. In the fourth quadrant, the y-values are negative, so y/r and y/x are negative; only x/r is positive, so only the Cosine is positive in QIV.

 

You're probably wondering why I capitalized the trig ratios and the word "All" in the preceding paragraph. I did that to explain this picture:

 

The letters in the quadrants stand for the initials of the trig ratios which are positive in that quadrant.

 

axis system, with 'A' in QI, 'S' in QII, 'T' in QIII, and 'C' in QIV

Some people remember the letters using the word "ACTS", but that's the reverse of normal (anti-clockwise) trigonometric order. Others remember the letters with the word "CAST", which is the normal rotational order but doesn't start in the usual (first-quadrant) starting place. To start in the usual spot and rotate in the usual direction, still others use the mnemonic "All Students Take Calculus" (which is a bit ironic when you're in a trig class). Use whichever method works best for you.

  • Find the values of the remaining trigometric ratios, given that cos(θ) = –8/17 and theta lies in QIII.
  • From the sign on the cosine value, I only know that the angle is in QII or QIII. That's why they had to give me that additional specification: so I'd know which of those quadrants I'm in.

     

    I'll start my work by drawing a picture
    of what I know so far:

     

    x- and y-axes drawn, terminal side in QIII, base x = -8, hypotenuse r = 17, theta labelled

    The Pythagorean Theorem gives me:

      172 = 82 + y2
      289 = 64 + y2

      225 = y2

      ±15 = y

    Since I'm in QIII, I'm below the x-axis, so y is negative. I'll take the negative solution to the equation, and I'll add this to my picture:

    Now I can read off the values from my picture:

     

     

     

    same picture, but with y = -15 added to height line of right triangle

      sin(theta) = -15/17, tan(theta) = 15/8, csc(theta) = -17/15, sec(theta) = -17/8, cot(theta) = 8/15


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