# Secant smaller than hypotenuse?

• Feb 13th 2010, 12:15 AM
pinkperil
Secant smaller than hypotenuse?
I am having an issue with understanding the concept of a secant and to some extent a tangent. Yes all the basic trig functions are ratios, but the sine ratio represents an angle. I have seen diagrams representing a secant as a line, but it is something more abstract. I have divided the hypotenuse by the adjacent line, but 4.47 / 4 = 1.1... so obviously the value of 1.1 is much smaller than 4.47. Apparently, the secant line is longer than the hypotenuse in all the diagrams that I have seen ...so I am at a lost as to how a secant translates into the longer line.
• Feb 13th 2010, 01:00 AM
Hello pinkperil

Welcome to Math Help Forum!
Quote:

Originally Posted by pinkperil
I am having an issue with understanding the concept of a secant and to some extent a tangent. Yes all the basic trig functions are ratios, but the sine ratio represents an angle. I have seen diagrams representing a secant as a line, but it is something more abstract. I have divided the hypotenuse by the adjacent line, but 4.47 / 4 = 1.1... so obviously the value of 1.1 is much smaller than 4.47. Apparently, the secant line is longer than the hypotenuse in all the diagrams that I have seen ...so I am at a lost as to how a secant translates into the longer line.

Don't worry about it. I've been teaching trigonometry for 45 years, and I've never yet had to use any geometrical property of the secant of an angle. Just learn that:
$\displaystyle \sec \theta = \frac{1}{\cos\theta}=\frac{\text{hypotenuse}}{\tex t{adjacent}}$
• Feb 15th 2010, 09:58 PM
pinkperil
Thanks for the reply grandad. It took a while, but I figured what my problem was. I was reading secant theta as the angle of secant instead of the secant OF theta. I was working on conversion between degrees and the trig ratios, so I was reading the ratios literally. It's not easy telling which context is used, but now that I am aware of the two different meanings, I should be able to tell. So anyhow, I figured out how to calculate the secant in a unit circle:

I let the hypotenuse equal the radius and divide it by the adjacent. Then once I get the angle, I let the adjacent equal the radius. Then I do the usual inverse operation to get the length of the hypotenuse that cuts through the circle, which of course is plainly the secant.
• Feb 16th 2010, 12:22 AM
If I understand you correctly, you're using the fact that if the length of the side adjacent to the angle $\displaystyle \theta$ in a right-angled triangle is $\displaystyle 1$ unit, then the length of the hypotenuse is $\displaystyle \sec\theta$.