Textbook Question
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
cos 75°
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.26
Textbook Question
For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.
cos(-55°)
Verified step by step guidance1
Recognize that the cosine function is an even function, which means that \( \cos(-x) = \cos(x) \).
Apply the even function property to the given expression: \( \cos(-55°) = \cos(55°) \).
Identify that \( \cos(55°) \) is the expression from Column II that matches the value of \( \cos(-55°) \).
Verify that no further simplification or identity is needed since the expression is already in its simplest form.
Conclude that the expression from Column II with the same value as \( \cos(-55°) \) is \( \cos(55°) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even-Odd Identities
Even-odd identities in trigonometry describe how the cosine and sine functions behave with negative angles. Specifically, the cosine function is even, meaning that cos(-θ) = cos(θ). This property allows us to simplify expressions involving negative angles by replacing them with their positive counterparts.
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Trigonometric Values of Common Angles
Understanding the trigonometric values of common angles, such as 0°, 30°, 45°, 60°, and 90°, is essential for solving trigonometric problems. These values are often used in conjunction with identities to simplify expressions and find equivalent forms, making it easier to match expressions from different columns.
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Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. They include fundamental identities like the Pythagorean identity, reciprocal identities, and co-function identities. These identities are crucial for transforming and simplifying trigonometric expressions, allowing for the identification of equivalent expressions.
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