Textbook Question
Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(cos x sin 2x)/1 + cos 2x)
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
Problem 5.18
Textbook Question
The half-angle identity
tan A/2 = ± √[(1 - cosA)/(1 + cos A)]
can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity
tan A/2 = sin A/(1 + cos A)
can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)
Verified step by step guidance1
Start by using the half-angle identity \( \tan \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{1 + \cos A}} \) to express \( \tan 22.5^\circ \).
Substitute \( A = 45^\circ \) into the identity, since \( 22.5^\circ = \frac{45^\circ}{2} \).
Calculate \( \cos 45^\circ \), which is \( \frac{\sqrt{2}}{2} \), and substitute it into the expression \( \tan 22.5^\circ = \sqrt{\frac{1 - \cos 45^\circ}{1 + \cos 45^\circ}} \).
Simplify the expression \( \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}} \) to show that it equals \( \sqrt{3 - 2\sqrt{2}} \).
Use the other half-angle identity \( \tan \frac{A}{2} = \frac{\sin A}{1 + \cos A} \) with \( A = 45^\circ \) to show that \( \tan 22.5^\circ = \sqrt{2} - 1 \), and verify that both expressions are equivalent by showing their squares are equal.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Half-Angle Identities
Half-angle identities are trigonometric formulas that express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the original angle. For tangent, the half-angle identity is given by tan(A/2) = ±√[(1 - cosA)/(1 + cosA)] or tan(A/2) = sinA/(1 + cosA). These identities are essential for simplifying expressions and solving problems involving angles that are halved.
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Trigonometric Equivalence
Trigonometric equivalence refers to the property that different trigonometric expressions can yield the same value under certain conditions. In this case, the two forms of the half-angle identity for tangent can be shown to be equivalent when applied to the specific angle of 22.5°. Understanding this concept allows for the manipulation and transformation of trigonometric expressions to demonstrate their equality.
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Square Root Properties
The square root property states that if a² = b², then a = b or a = -b, provided that a and b are real numbers. This property is crucial in the context of the problem, as it allows us to conclude that if two expressions for tan(22.5°) yield the same squared value, they must be equal in magnitude. This principle is used to validate the equivalence of the two different forms of the tangent half-angle identity.
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