Why is sec(theta) different than sqrt(1+tan(theta).^2) in MATLAB?

Hi all,

I just run into an odd problem; we all know the trigonometric identity . I use this identity in my code where I have a vector of angles of which I would like to calculate the secant of:

Psi = [0 320 40 280 80 240 120 200 160]*pi/180 ; % [rad]
x = [sec(Psi) ; sqrt(1 + tan(Psi).^2)] ;

>> x =

    1.0000    1.3054    1.3054    5.7588    5.7588   -2.0000   -2.0000   -1.0642   -1.0642
    1.0000    1.3054    1.3054    5.7588    5.7588    2.0000    2.0000    1.0642    1.0642

Above, I would expect both row vectors to be identical however it is visible that some numbers are returned as negative by the sec and postitive by the equivalent identity.

As a result I would like to ask: 1. why does this discrepancy exist to begin with 2. which one is correct?

Answer 1

[Walter Roberson]

That is not a correct statement of the trig identity. The trig identity states that but that does not mean that -- it means that

[Steven Lord]

Many of the entries in the the Pythagorean identities section on this Wikipedia page use ±.

Answer 2

[John D'Errico]

First, what you have written is NOT an identity. The identity is

sec(theta)^2 = 1 + tan(theta)^2

There IS a difference! You should remember that taking the square root is not valid there, because there can be a problem with the sign.

So this is not a problem in MATLAB, but a problem in your mathematics. Which one is correct? What is correct is to use the correct identity.

Answer 3

[Voss]

sqrt(x) returns the positive square root of a real number x. For example, sqrt(4) returns 2, but -2 is also a square root of 4.

sec(x) returns the secant of the angle x, which will be positive or negative depending on which quadrant x is in.

So to answer your questions: 1. the discrepancy is due to the sqrt function picking the positive square root. 2. they are both correct.

The identity in question is more correctly written as sec(x)^2 == 1 + tan(x)^2 (the square of both sides of the one you stated), which does not have the ambiguity due to the sqrt.