Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:αa. sin ------2𝝅 sec α = ﹣3 , ------ < α < 𝝅2
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
5:57 minutesProblem 65
Textbook Question
In Exercises 59–68, verify each identity.x sin xcot ------ = -----------------2 1 - cos x
Verified step by step guidance1
Step 1: Start by rewriting the given identity: \( \frac{x \sin x \cot \frac{x}{2}}{1 - \cos x} \).
Step 2: Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Apply this to \( \cot \frac{x}{2} \) to get \( \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}} \).
Step 3: Substitute \( \cot \frac{x}{2} \) in the identity: \( \frac{x \sin x \cdot \frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}}{1 - \cos x} \).
Step 4: Simplify the expression by multiplying: \( \frac{x \sin x \cos \frac{x}{2}}{\sin \frac{x}{2} (1 - \cos x)} \).
Step 5: Use trigonometric identities or algebraic manipulation to further simplify and 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 co-function identities. Understanding these identities is crucial for simplifying expressions and verifying equations in trigonometry.
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Cotangent Function
The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is important to understand how cotangent relates to sine and cosine, as this relationship is often used in trigonometric identities and simplifications. Recognizing how to manipulate these functions is key to solving problems involving cotangent.
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Sine and Cosine Functions
Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides. The sine function gives the ratio of the opposite side to the hypotenuse, while the cosine function gives the ratio of the adjacent side to the hypotenuse. Mastery of these functions is essential for working with trigonometric identities and solving equations involving angles.
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