Textbook Question
Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(1 - cos 2x)/sin 2x
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.4
Textbook Question
Determine whether the positive or negative square root should be selected.
sin (-10°) = ± √[(1 - cos (-20°))/2]
Verified step by step guidance1
Step 1: Understand the problem. We need to determine whether to use the positive or negative square root for the expression involving sine and cosine.
Step 2: Recall the identity for sine of a negative angle: \( \sin(-\theta) = -\sin(\theta) \). Therefore, \( \sin(-10^\circ) = -\sin(10^\circ) \).
Step 3: Use the cosine double angle identity: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). For \( \cos(-20^\circ) \), note that \( \cos(-\theta) = \cos(\theta) \), so \( \cos(-20^\circ) = \cos(20^\circ) \).
Step 4: Substitute \( \cos(-20^\circ) \) into the expression: \( \pm \sqrt{\frac{1 - \cos(20^\circ)}{2}} \).
Step 5: Since \( \sin(-10^\circ) = -\sin(10^\circ) \) is negative, select the negative square root for the expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sine Function and Its Properties
The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is periodic with a period of 360°, meaning sin(θ) = sin(θ + 360°n) for any integer n. Additionally, sine is an odd function, which implies that sin(-θ) = -sin(θ). This property is crucial for evaluating sin(-10°) in the given equation.
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Cosine Function and Its Properties
The cosine function, represented as cos(θ), is the ratio of the length of the adjacent side to the hypotenuse in a right triangle. Unlike sine, cosine is an even function, meaning cos(-θ) = cos(θ). This property is important when calculating cos(-20°) in the equation, as it simplifies the expression and helps determine the value needed for the sine calculation.
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Square Roots and Their Sign
When dealing with square roots, it is essential to consider both the positive and negative roots, as both can satisfy the equation x² = a. In trigonometric contexts, the choice between the positive or negative root often depends on the specific quadrant in which the angle lies. For instance, since sin(-10°) is negative, it is important to select the negative square root when evaluating the expression √[(1 - cos(-20°))/2] to maintain consistency with the sine function's value.
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