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Question: How do you find the exact value of \(2\cos x - \sec x = 0\) in the interval \(0 \leqslant x < 360?\)...

How do you find the exact value of 2cos⁡x−sec⁡x=02\cos x - \sec x = 0 in the interval 0⩽x<360?0 \leqslant x < 360?

Explanation

Solution

In this question, we are going to solve the given equation for the given interval.
First we are going to write the secant xx by the reciprocal identity and then multiplying by cos⁡x\cos x on both sides.
And then simplifying the equation we get the result and then we can get the solution from the given interval.

Formula used: The reciprocal identity is written as
sec⁡x=1cos⁡x\sec x = \dfrac{1}{{\cos x}}

Complete step by step solution:
In this question, we are going to solve the given equation by the given interval.
First write the given equation and mark it as (1)\left( 1 \right)
2cos⁡x−sec⁡x=0………....(1)2\cos x - \sec x = 0………....\left( 1 \right)
Now by applying the reciprocal identity to the cosecant xx in equation (1)\left( 1 \right) we get,
⇒2cos⁡x−1cos⁡x=0\Rightarrow 2\cos x - \dfrac{1}{{\cos x}} = 0
Multiplying cos⁡x\cos x on both sides of the equation we get,
⇒2cos⁡2x−cos⁡xcos⁡x=0\Rightarrow 2{\cos ^2}x - \dfrac{{\cos x}}{{\cos x}} = 0
Simplify the above equation we get,
⇒2cos⁡2x−1=0\Rightarrow 2{\cos ^2}x - 1 = 0
On rewriting we get,
⇒2cos⁡2x=1\Rightarrow 2{\cos ^2}x = 1
Let us divide the term and we get,
⇒cos⁡2x=12\Rightarrow {\cos ^2}x = \dfrac{1}{2}
Taking square on both side we get,
⇒cos⁡x=±12\Rightarrow \cos x = \pm \dfrac{1}{{\sqrt 2 }}
Hence we can get the values of xx from the unit circle.
x=45∘,135∘,225∘,315∘x = {45^ \circ },{135^ \circ },{225^ \circ },{315^ \circ }

Therefore the solution for the trigonometric equation 2cos⁡x−sec⁡x=02\cos x - \sec x = 0 is 45∘,135∘,225∘,315∘{45^ \circ },{135^ \circ },{225^ \circ },{315^ \circ }

Note: Solving the trigonometric equation is a tricky work that often leads to errors and mistakes. Therefore, answers should be carefully checked. After solving, you can check the answers by using a graph.
The unit circle or trigonometric circle as it is also known is useful to know because it lets us easily calculate the cosine, sine, and tangent of any angle between 00 and 360360 degrees.
Trigonometry is also helpful to measure the height of the mountain, to find the distance of long rivers, etc. its applications are in various fields like oceanography, astronomy, navigation, electronics, physical sciences etc.