Find one value of θ or x that satisfies each of the following.
sec x = csc (2π/3)

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Recall the definitions of the secant and cosecant functions: \(\sec x = \frac{1}{\cos x}\) and \(\csc y = \frac{1}{\sin y}\).

Rewrite the given equation \(\sec x = \csc \left( \frac{2\pi}{3} \right)\) as \(\frac{1}{\cos x} = \frac{1}{\sin \left( \frac{2\pi}{3} \right)}\).

Simplify the equation to \(\cos x = \sin \left( \frac{2\pi}{3} \right)\) by taking the reciprocal of both sides.

Evaluate \(\sin \left( \frac{2\pi}{3} \right)\) using the unit circle or known sine values for special angles.

Set \(\cos x\) equal to the value found in the previous step and solve for \(x\) by considering the general solutions for cosine, which are \(x = \pm \arccos(\text{value}) + 2k\pi\), where \(k\) is any integer.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Reciprocal Trigonometric Functions

Secant (sec) and cosecant (csc) are reciprocal functions of cosine and sine, respectively. Specifically, sec x = 1/cos x and csc x = 1/sin x. Understanding these relationships helps convert the given equation into a more manageable form involving sine and cosine.

Evaluating Trigonometric Functions at Special Angles

The angle 2π/3 is a special angle in the unit circle, located in the second quadrant. Knowing the exact values of sine and cosine at this angle (sin 2π/3 = √3/2, cos 2π/3 = -1/2) allows precise evaluation of csc(2π/3), which is needed to solve the equation.

Solving Trigonometric Equations

To find values of x satisfying sec x = csc(2π/3), rewrite the equation in terms of sine and cosine, then solve for x. This involves understanding the periodicity and domain of trigonometric functions to find valid solutions within a given interval.

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