Question

What is the value of $\sec \left( {{180}^{\circ }} \right)?$

Answer 1

Hint: We need to find the value of $\sec \left( {{180}^{\circ }} \right)$ . We start to solve the problem by finding out the value of $\sec \left( {{180}^{\circ }} \right)$ in terms of the cosine trigonometric function. Then, we need to simplify the expression to get the desired result.

Complete step by step solution:
We are asked to find the value of $\sec \left( {{180}^{\circ }} \right)$. We will be solving the given question using the basic formulae in trigonometry.
Secant is the trigonometric function of any specified angle that is used in the context of a right angle. It is usually defined as the ratio of the length of the hypotenuse to the length of the side adjacent to an angle of the right-angle triangle.
Cosine is the trigonometric function of any specified angle that is used in the context of a right angle.
It is usually defined as the ratio of the length of the side adjacent to an angle to the length of the hypotenuse of the right-angle triangle.
According to the question,
We need to find the value of $\sec \left( {{180}^{\circ }} \right)$
$\Rightarrow \sec \left( {{180}^{\circ }} \right)$
From trigonometry,
We know that the secant function is the reciprocal of the cosine function.
Writing the same in the form of the equation, we get,
$\Rightarrow \sec x=\dfrac{1}{\cos x}$
Substituting the formula in the above expression, we get,
$\Rightarrow \sec \left( {{180}^{\circ }} \right)=\dfrac{1}{\cos \left( {{180}^{\circ }} \right)}$
Now, we need to find the value of $\cos \left( {{180}^{\circ }} \right)$
The value of the trigonometric function $\cos \left( {{180}^{\circ }} \right)$ can be also written as follows,
$\Rightarrow \cos \left( {{180}^{\circ }} \right)=\cos \left( {{90}^{\circ }}+{{90}^{\circ }} \right)$
The angle $\left( {{90}^{\circ }}+{{90}^{\circ }} \right)$ lies in the ${{2}^{nd}}$ quadrant and only the sine and cosecant functions are positive in the ${{2}^{nd}}$quadrant.
So, the value of $\cos \left( {{90}^{\circ }}+{{90}^{\circ }} \right)$ is negative in the ${{2}^{nd}}$ quadrant.
The cosine function changes to the sine function when is the angle is 90 or 270.
Following the same, we get,
$\Rightarrow \cos \left( {{180}^{\circ }} \right)=-\sin \left( {{90}^{\circ }} \right)$
From trigonometry, we know that $\sin \left( {{90}^{\circ }} \right)=1$
Substituting the same, we get,
$\Rightarrow \cos \left( {{180}^{\circ }} \right)=-1$
Lastly, we need to substitute the value of $\cos \left( {{180}^{\circ }} \right)$ in the equation $\sec \left( {{180}^{\circ }} \right)=\dfrac{1}{\cos \left( {{180}^{\circ }} \right)}$
Substituting the value of $\cos \left( {{180}^{\circ }} \right)$ in the equation, we get,
$\Rightarrow \sec \left( {{180}^{\circ }} \right)=\dfrac{1}{\left( -1 \right)}$
$\therefore \sec \left( {{180}^{\circ }} \right)=-1$

Hence, the value of $\sec \left( {{180}^{\circ }} \right)$ is equal to -1.

Note: We must remember that the cosine function is positive in the ${{1}^{st}}$ and ${{4}^{th}}$ quadrants and is negative in the ${{2}^{nd}}$ and ${{3}^{rd}}$ quadrants. The cosine function changes to the sine function when the angle is 90 or 270.