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Question: How do you solve \(2+\sec x=0\) and find all solutions in the interval \({{0}^{\circ }}\le x<{{360}^...

How do you solve 2+sec⁡x=02+\sec x=0 and find all solutions in the interval 0∘≤x<360∘{{0}^{\circ }}\le x<{{360}^{\circ }} ?

Explanation

Solution

In order to find solution to this trigonometric equation, we will use trigonometric functions and in order to find the solution in the interval of xx within the range, we will subtract xx from 360360, that is first by using trigonometric function properties we will find first value of xx and then we will subtract that value from 360360 to get another value of xx.

Complete step by step answer:
In geometry, trigonometric functions are the functions that are used to find the unknown angle or side of a right-angled triangle.
There are three basic trigonometry functions: Sine, Cosine and Tangent.
The angles of sine, cosine, and tangent are the primary classification of functions of trigonometry and the three functions which are cotangent, secant and cosecant can be derived from the primary functions.
So Basically, the other three functions are often used as compare to the primary trigonometric functions.
Now coming to our question, we have:
⇒2+sec⁡x=0\Rightarrow 2+\sec x=0
Now, we will use trigonometric function properties, that is sec⁡x=1cos⁡x\sec x=\dfrac{1}{\cos x} :
Therefore, on implying and shifting 22 on the other side, we get our equation as:
⇒1cos⁡x=−2\Rightarrow \dfrac{1}{\cos x}=-2
On re-rearranging the terms in the expression, we get:
⇒cos⁡x=−12\Rightarrow \cos x=-\dfrac{1}{2}
Now, on taking cosine on right-hand side, we get:
⇒x=cos⁡−1(−12)\Rightarrow x={{\cos }^{-1}}\left( -\dfrac{1}{2} \right)
That is
⇒x=120∘\Rightarrow x={{120}^{\circ }} since cos⁡−1(−12)=120∘{{\cos }^{-1}}\left( -\dfrac{1}{2} \right)={{120}^{\circ }}
Now in order to find the other value of xx within the range, we will subtract xx from 360360.
Therefore, we get:
⇒x=360−120\Rightarrow x=360-120
On simplifying, we get:
⇒x=240∘\Rightarrow x={{240}^{\circ }}
Therefore, the solutions in the interval 0∘≤x<360∘{{0}^{\circ }}\le x<{{360}^{\circ }} is:
⇒x=120∘,240∘\Rightarrow x={{120}^{\circ }},{{240}^{\circ }}, which is the required answer.

Note: Trigonometric functions are also known as a Circular Functions can be simply defined as the functions of an angle of a triangle.
It means that the relationship between the angles and sides of a triangle are given by these trigonometric functions.
The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant.