Textbook Question
Write each function as an expression involving functions of θ or x alone. See Example 2.
cos(45° - θ)
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6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.39
Textbook Question
Find one value of θ or x that satisfies each of the following.
sec x = csc (2π/3)
Verified step by step guidance1
Recognize that \( \sec x = \frac{1}{\cos x} \) and \( \csc \theta = \frac{1}{\sin \theta} \).
Calculate \( \csc \left(\frac{2\pi}{3}\right) \) by finding \( \sin \left(\frac{2\pi}{3}\right) \).
Recall that \( \sin \left(\frac{2\pi}{3}\right) = \sin \left(\pi - \frac{\pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) \).
Use the known value \( \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) to find \( \csc \left(\frac{2\pi}{3}\right) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).
Set \( \sec x = \frac{2}{\sqrt{3}} \) and solve for \( x \) by finding \( \cos x = \frac{\sqrt{3}}{2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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