Equivalence of Definitions of Secant of Angle

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Theorem

Let $\theta$ be an angle.


The following definitions of the concept of secant are equivalent:

Definition from Triangle

SineCosine.png

In the above right triangle, we are concerned about the angle $\theta$.

The secant of $\angle \theta$ is defined as being $\dfrac{\text{Hypotenuse}} {\text{Adjacent}}$.

Definition from Circle

Consider a unit circle $C$ whose center is at the origin of a cartesian coordinate plane.


SecantFirstQuadrant.png


Let $P$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.

Let a tangent line be drawn to touch $C$ at $A = \left({1, 0}\right)$.

Let $OP$ be produced to meet this tangent line at $B$.


Then the secant of $\theta$ is defined as the length of $OB$.


Proof

Definition from Triangle implies Definition from Circle

Let $\sec \theta$ be defined as $\dfrac {\text{Hypotenuse}} {\text{Adjacent}}$ in a right triangle.

Consider the triangle $\triangle OAB$.

By construction, $\angle OAB$ is a right angle.


Thus:

\(\displaystyle \sec \theta\) \(=\) \(\displaystyle \frac {OB} {OA}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {OB} 1\) as $OA$ is the radius of the unit circle
\(\displaystyle \) \(=\) \(\displaystyle OB\)

That is:

$\sec \theta = OB$

$\Box$


Definition from Circle implies Definition from Triangle

Let $\sec \theta$ be defined as the length of $OB$ in the triangle $\triangle OAB$.

Compare $\triangle OAB$ with $\triangle ABC$ in the diagram above.

We have that:

$\angle CAB = \angle BOA = \theta$
$\angle ABC = \angle OAB$ which is a right angle

Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAB$ and $\triangle ABC$ are similar.

By definition of similarity:

\(\displaystyle \frac {\text{Hypotenuse} } {\text{Adjacent} }\) \(=\) \(\displaystyle \frac {AC} {AB}\) by definition
\(\displaystyle \) \(=\) \(\displaystyle \frac {OB} {OA}\) by definition of similarity
\(\displaystyle \) \(=\) \(\displaystyle OB\) as $OA$ is the radius of the unit circle
\(\displaystyle \) \(=\) \(\displaystyle \sec \theta\) by definition

That is:

$\dfrac {\text{Hypotenuse} } {\text{Adjacent} } = \sec \theta$

$\blacksquare$