Solving precalc problem without calculator

[bfpri]

The terminal side of an angle theta in standard position passes through (-2,-3). What is sec theta?

How would you solve this problem without a calculator?

Thanks.

 

[symbolipoint]

One approach to start with is draw this angle on a cartesian coordinate system, maybe including in the form of a triangle; you may need to express something using pythagorean theorem. Remember that 1/(cosine) = secant.

 

[HallsofIvy]

you could set up a triangle having (0,0) as one vertex, (-2, -3) as another and (-2, 0) as the third. What are the lengths of the three sides? What is sec of the angle at the origin (which is what you must mean by "theta"- you didn't specify what "theta" was!). Remember that these are "signed" lengths. Distances measured downward or to the left are negative.

Even simpler would be to use the "circular" definition of sine and cosine: Draw a unit circle. Starting at (1, 0) measure around the circumference a distance t. The coordinates of the end point are, by definition, (cos t, sin t). Now, (-2, -3) does not lie on the unit circle but it is easy to find a point on that line that does (use "similar triangles"). What is the x coordinate of that point? What is 1/cosine?