Textbook Question
In Exercises 15–22, write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2 sin 15° cos 15°
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6. Trigonometric Identities and More Equations
Double Angle Identities
3:33 minutesProblem 3.3.53
Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. 2sin(θ/2)cos(θ/2)
Verified step by step guidance1
Identify the given trigonometric expressions: \( 2 \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \). The problem likely involves finding exact values related to these half-angle expressions.
Recall the half-angle formulas for sine and cosine:
\[ \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \quad \text{and} \quad \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \]
Determine the correct sign based on the quadrant where \( \frac{\theta}{2} \) lies.
Substitute the known value of \( \cos \theta \) from the figure or given information into the half-angle formulas to express \( \sin \frac{\theta}{2} \) and \( \cos \frac{\theta}{2} \) in exact radical form.
Multiply \( \sin \frac{\theta}{2} \) by 2 as indicated in the expression \( 2 \sin \frac{\theta}{2} \) to simplify or rewrite the expression accordingly.
Combine the expressions or use Pythagorean identities if needed to find the exact values of the trigonometric functions requested, ensuring all steps use exact values (no decimal approximations).
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions and Their Definitions
Trigonometric functions like sine and cosine relate the angles of a triangle to the ratios of its sides. Specifically, sine of an angle is the ratio of the opposite side to the hypotenuse, while cosine is the ratio of the adjacent side to the hypotenuse. Understanding these definitions is essential for evaluating trigonometric expressions.
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Angle Halving and the Use of θ/2
The expression involves trigonometric functions of half an angle (θ/2). Recognizing how to work with half-angle values is important, as it often requires applying half-angle identities or understanding how the angle relates to the original θ to find exact values.
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Half-Angle Identities
Half-angle identities provide formulas to find sine and cosine of half an angle using the cosine or sine of the original angle θ. For example, sin(θ/2) = ±√((1 - cos θ)/2) and cos(θ/2) = ±√((1 + cos θ)/2). These identities are crucial for finding exact trigonometric values when given θ.
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