Question

How do I get the exact value of $\sec 225{}^\circ $.

Answer 1

Hint: In this problem we need to calculate the value of trigonometric ratio $\sec $ for a given angle. For this we will convert the trigonometric ratio $\sec x$ in terms of $\cos x$ by using the known basic formula which is $\sec x=\dfrac{1}{\cos x}$. From this we can say that the value of $\sec 225{}^\circ $ depends on the value of $\cos 225{}^\circ $. Now we will write the given angle $225{}^\circ $ as $180{}^\circ +45{}^\circ $. Substitute this in the value of $\cos 225{}^\circ $ . Now we will use the trigonometric formula $\cos \left( x+y \right)=\cos x.\cos y-\sin x.\sin y$. After applying this formula, we will substitute the known values for the trigonometric ratios and simplify to get the required result.

Complete step by step answer:
Given $\sec 225{}^\circ $.
We know that $\sec x=\dfrac{1}{\cos x}$, then the value of $\sec 225{}^\circ $ will be
$\sec 225{}^\circ =\dfrac{1}{\cos 225{}^\circ }$
Considering the term $\cos 225{}^\circ $ which is in the numerator.
We can write the given value $225{}^\circ $ as $225{}^\circ =180{}^\circ +45{}^\circ $. Substituting this value in $\cos 225{}^\circ $, then we will get
$\cos 225{}^\circ =\cos \left( 180{}^\circ +45{}^\circ \right)$
We have the trigonometric formula $\cos \left( x+y \right)=\cos x.\cos y-\sin x.\sin y$. Applying this formula in the above equation, then we will get
$\Rightarrow \cos 225{}^\circ =\cos 180{}^\circ .\cos 45{}^\circ -\sin 180{}^\circ .\sin 45{}^\circ $
We have the values $\cos 180{}^\circ =-1$, $\sin 180{}^\circ =0$, $\sin 45{}^\circ =\cos 45{}^\circ =\dfrac{1}{\sqrt{2}}$. Substituting these values in the above equation, then we will have
$\begin{align}
  & \Rightarrow \cos 225{}^\circ =-1\times \dfrac{1}{\sqrt{2}}-0\times \dfrac{1}{\sqrt{2}} \\
 & \Rightarrow \cos 225{}^\circ =-\dfrac{1}{\sqrt{2}} \\
\end{align}$
From the value of $\cos 225{}^\circ $, the value of $\sec 225{}^\circ $ will be
$\Rightarrow \sec 225{}^\circ =\dfrac{1}{\cos 225{}^\circ }$
Substituting the value of $\cos 225{}^\circ $ in the above equation, then we will have
$\begin{align}
  & \Rightarrow \sec 225{}^\circ =\dfrac{1}{\left( -\dfrac{1}{\sqrt{2}} \right)} \\
 & \therefore \sec 225{}^\circ =-\sqrt{2} \\
\end{align}$

Note: We can also simply calculate the value of $\sec 225{}^\circ $ without converting it into $\cos 225{}^\circ $. We can write $225{}^\circ =180{}^\circ +45{}^\circ $, then we will have
$\sec 225{}^\circ =\sec \left( 180{}^\circ +45{}^\circ \right)$
From the above equation, we can say that the angle lies in the third quadrant and we know that there is no trigonometric ratio change for angles $180{}^\circ +\theta $ and in third quadrant $\tan ,\cot $ are the positive ratios only. Then we will get
$\Rightarrow \sec 225{}^\circ =-\sec 45{}^\circ $
We know that $\sec 45{}^\circ =\sqrt{2}$, then
$\therefore \sec 225{}^\circ =-\sqrt{2}$
From both the methods we got the same result.