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

Verified step by step guidance

1. Recognize that \( \sec x = \frac{1}{\cos x} \) and \( \csc \theta = \frac{1}{\sin \theta} \).
2. Calculate \( \csc \left(\frac{2\pi}{3}\right) \) by finding \( \sin \left(\frac{2\pi}{3}\right) \).
3. Recall that \( \sin \left(\frac{2\pi}{3}\right) = \sin \left(\pi - \frac{\pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) \).
4. Use the known value \( \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) to find \( \csc \left(\frac{2\pi}{3}\right) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).
5. Set \( \sec x = \frac{2}{\sqrt{3}} \) and solve for \( x \) by finding \( \cos x = \frac{\sqrt{3}}{2} \).

Key Concepts

Secant and Cosecant Functions

The secant function, denoted as sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). Understanding these functions is crucial for solving equations involving them, as it allows for the manipulation and transformation of trigonometric identities.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identities, reciprocal identities, and co-function identities. These identities are essential for simplifying expressions and solving equations in trigonometry, such as transforming sec(x) and csc(2π/3) into more manageable forms.

Angle Measurement and Reference Angles

In trigonometry, angles can be measured in degrees or radians, with 2π radians equivalent to 360 degrees. The reference angle is the acute angle formed by the terminal side of an angle and the x-axis, which helps in determining the values of trigonometric functions in different quadrants. Understanding how to find and use reference angles is important for solving equations like sec(x) = csc(2π/3) effectively.