Question

How do you find the exact value of \[2\sec x=2+\sec x\] in the interval \[0\le x<360\]?

Answer 1

Hint: In the given question, we have been asked to find the exact value of the given trigonometric expression and the interval for the trigonometric function is also given. In order to find the exact value, first we need to simplify the equation and write it in the form of cosine function. Later considering the given interval, from the trigonometric unit circle by using trigonometric ratios table we will get to know that exact value of the given trigonometric equation.

Complete step by step solution:
We have given that,
\[2\sec x=2+\sec x\]
Rewritten the above trigonometric expression as,
\[2\sec x-\sec x=2\]
Simplifying the above given trigonometric expression, we get
\[\sec x=2\]
Using the trigonometric identity, i.e. \[\sec x=\dfrac{1}{\cos x}\]
Applying this trigonometric identity in the above expression, we have
\[\dfrac{1}{\cos x}=2\]
Simplifying the above expression, we obtained
\[\cos x=\dfrac{1}{2}\]
As we know that,
Using the trigonometric ratios table, \[\cos \left( {{60}^{0}} \right)=\dfrac{1}{2}\]
From the trigonometric unit circle;
\[\cos \left( {{360}^{0}}-{{60}^{0}} \right)=\dfrac{1}{2}\]
Thus,
We have two values of ‘x’ i.e.
\[x={{60}^{0}}or\ x={{300}^{0}}\]
And in degrees;
\[x=\dfrac{\pi }{3}\ or\ \dfrac{5\pi }{3}\]
Hence, it is the required answer.

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.