Math Doubts

$\sec{(0^°)}$ Proof

According to the trigonometric mathematics, the secant of zero degrees is equal to one.

$\sec{(0^°)} \,=\, 1$

The secant of angle zero degrees value can be derived exactly in three different mathematical approaches. One of them is related to trigonometry and the remaining two methods are related to geometry. Now, let’s learn how to prove the value for secant of angle zero radian from each method.

Fundamental method

A property between hypotenuse and adjacent side in a zero degree right triangle is considered for proving the exact value of secant of angle zero grades in this geometric method. For proving it, let’s imagine a right triangle that contains zero degrees angle. Here, $\Delta QPR$ is an understandable example for right angled triangle with zero angle.

properties of right angled triangle when angle is 0 degrees

As per the definition of the secant function, it can be expressed in mathematical form by writing it in ratio of lengths of the sides. In fact, the secant function is written as $\sec{(0^°)}$ when the angle of a right triangle equals to zero degrees.

$\implies$ $\sec{(0^°)}$ $\,=\,$ $\dfrac{Length \, of \, Hypotenuse}{Length \, of \, Adjacent \, side}$

$\implies$ $\sec{(0^°)}$ $\,=\,$ $\dfrac{PQ}{PR}$

The lengths of both adjacent side and hypotenuse are equal in a zero degree right triangle. It means $PQ \,=\, PR$.

$\require{cancel} \implies$ $\sec{(0^°)}$ $\,=\,$ $\dfrac{\cancel{PR}}{\cancel{PR}}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\sec{(0^°)}$ $\,=\,$ $1$

Experimental method

The secant of angle zero degrees can also be derived practically by constructing a zero degree right angled triangle. In general, it can be constructed by the geometric tools. Now, let’s start the procedure for constructing a right triangle with zero degrees angle.

right angled triangle with 0 degrees angle for sec 0 degrees value
  1. From point $M$, draw a horizontal straight line by a ruler.
  2. Now, coincide the middle point with point $M$ and also coincide its right side baseline with horizontal line. After that, draw a straight line with zero degree angle from point $M$ and it is drawn actually over the horizontal line.
  3. Take a compass and Set the distance between pencil lead point and needle point to any length (for example $6$ centimetres). Later, draw an arc on zero degrees line and it intersects the zero angle straight line at point $N$.
  4. Finally, use set square and draw a perpendicular line to horizontal line from point $N$ and it intersects the horizontal line at point $O$. Thus, we can construct $\Delta NMO$. In fact, it is more complicated to draw a perpendicular line to horizontal line and it is impossible because the points $N$ and $O$ are located at same position in the plane.
right angled triangle with 0 degrees angle for sec 0 degrees value

In $\Delta NMO$, the adjacent side, hypotenuse and opposite side are $\overline{MO}$, $\overline{MN}$ and $\overline{NO}$ respectively. $MO = 6\,cm$, $MN = 6\,cm$ and $NO = 0\,cm$. Now, it is time to evaluate the secant of angle zero degrees by calculating the ratio of lengths of its corresponding sides.

$\sec{(0^°)} \,=\, \dfrac{MN}{MO}$

$\implies$ $\sec{(0^°)} \,=\, \dfrac{6}{6}$

$\require{cancel} \implies$ $\sec{(0^°)} \,=\, \dfrac{\cancel{6}}{\cancel{6}}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\sec{(0^°)} \,=\, 1$

Trigonometric method

The exact value for secant of angle zero degrees can also be derived from the fundamentals of trigonometry. According to the reciprocal trigonometric identity, the secant function can be expressed as reciprocal of the cosine function.

$\sec{\theta} \,=\, \dfrac{1}{\cos{\theta}}$

In this case, the angle $\theta$ is equal to zero degrees.

$\implies$ $\sec{(0^°)} \,=\, \dfrac{1}{\cos{(0^°)}}$

We know that the value of cos of zero degrees is equal to one.

$\implies$ $\sec{(0^°)} \,=\, \dfrac{1}{1}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\sec{(0^°)} \,=\, 1$

These are the possible methods to prove that secant of angle zero degrees is equal to one in mathematics.

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