# SAT Practice Questions: Sec, Csc, and Cot Questions

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:

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

## Practice questions

**Based on this figure,**

**what is the cosecant of angle theta**?

**In triangle***ABC*where*C*= 90 degrees and the hypotenuse is 5, csc*A*= 5/3. What is sec*B*?

## Answers and explanations

**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

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

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.**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.

Next, cos*B*is adjacent over hypotenuse, which is 3/5. Secant is the inverse of cosine, so sec*B*= 5/3.