SAT Practice Questions: Sec, Csc, and Cot Questions

By Ron Woldoff

Along with the trigonometric ratios represented by SOH CAH TOA, the SAT Math exam may have a question where you’ll have to work with secants, cosecants, or cotangents. Fortunately, it’s just a matter of finding the reciprocal of the sine, cosine, or tangent:

SAT1001_eq1301

The following practice questions ask you to do some reciprocal work with a unit circle and a right triangle.

Practice questions

  1. Based on this figure,
    sat1001_cosecant
    what is the cosecant of angle theta?
    SAT1001_eq1302
  2. In triangle ABC where C = 90 degrees and the hypotenuse is 5, csc A = 5/3. What is sec B?
    SAT1001_eq1303

Answers and explanations

  1. The correct answer is Choice (D).
    Draw a line from the labeled point to the x-axis, making a right triangle. If two of the sides are 1 and
    SAT1001_eq1304
    the third side is 2, and this is a 30-60-90 triangle, with angles measuring 30, 60, and 90 degrees. The smaller acute angle is 30 degrees, and
    SAT1001_eq1305
    which is supplementary, is 150 degrees.
    Cosecant is the reciprocal of sine, so start by finding the sine. Supplementary angles have the same sine, so you can measure the sine of the 30-degree angle. Using SOH CAH TOA, sine is opposite over hypotenuse. From the 30-degree angle, the side opposite is 1 and the hypotenuse is 2, for a sine of 1/2. Take the reciprocal for a cosecant of 2.
  2. The correct answer is Choice (A).
    Using SOH CAH TOA, sin A is opposite over hypotenuse; these sides are in the ratio of 3 to 5, respectively. Cosecant is the inverse of sine, so 3/5 becomes 5/3.
    sat1001-Cosecanta
    Next, cos B is adjacent over hypotenuse, which is 3/5. Secant is the inverse of cosine, so sec B = 5/3.