Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:αc. tan ------24tan α = ------ , 180° < α < 270°3
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
4:10 minutesProblem 62
Textbook Question
In Exercises 59–68, verify each identity.θ sec θ + 1cos² ------ = ---------------------2 2 sec θ
Verified step by step guidance1
Step 1: Start by rewriting the left-hand side of the identity. The expression is \( \frac{\cos^2 \theta}{2} \).
Step 2: Rewrite the right-hand side of the identity. The expression is \( \frac{\sec \theta + 1}{2 \sec \theta} \).
Step 3: Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Substitute this into the right-hand side to get \( \frac{\frac{1}{\cos \theta} + 1}{2 \cdot \frac{1}{\cos \theta}} \).
Step 4: Simplify the right-hand side by finding a common denominator for the numerator: \( \frac{1 + \cos \theta}{\cos \theta} \).
Step 5: Simplify the entire expression on the right-hand side: \( \frac{1 + \cos \theta}{2} \). Compare this with the left-hand side to verify the identity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Common identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is crucial for simplifying expressions and verifying equations in trigonometry.
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Secant Function
The secant function, denoted as sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). It is important to recognize how secant relates to cosine, as this relationship is often used in trigonometric identities and simplifications. Knowing how to manipulate secant in terms of cosine is essential for solving problems involving these functions.
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Simplifying Trigonometric Expressions
Simplifying trigonometric expressions involves using identities and algebraic techniques to rewrite expressions in a more manageable form. This process often includes factoring, combining like terms, and substituting equivalent expressions. Mastery of simplification techniques is vital for verifying identities and solving trigonometric equations effectively.
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