Textbook Question
In Exercises 43–44, express each product as a sum or difference. sin 6x sin 4x
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6. Trigonometric Identities and More Equations
Solving Trigonometric Equations Using Identities
4:27 minutesProblem 3.3.55a
Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:
a. sin(α/2)
tan α = 4/3, 180° < α < 270°
Verified step by step guidance1
Identify the quadrant where the angle \( \alpha \) lies. Since \( 180^\circ < \alpha < 270^\circ \), \( \alpha \) is in the third quadrant, where both sine and cosine values are negative, but tangent is positive.
Recall the given value: \( \tan \alpha = \frac{4}{3} \). Use the identity \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) to relate sine and cosine.
Set \( \sin \alpha = 4k \) and \( \cos \alpha = 3k \) for some constant \( k \), because the ratio of sine to cosine must be \( \frac{4}{3} \). Since \( \alpha \) is in the third quadrant, both sine and cosine are negative, so \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \).
Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \):
\[
(-4k)^2 + (-3k)^2 = 1 \\
16k^2 + 9k^2 = 1 \\
25k^2 = 1 \\
k^2 = \frac{1}{25} \\
k = \frac{1}{5}
\]
Substitute \( k = \frac{1}{5} \) back into \( \sin \alpha = -4k \) to find \( \sin \alpha \):
\[
\sin \alpha = -4 \times \frac{1}{5} = -\frac{4}{5}
\]
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Ratios and Their Signs in Quadrants
Trigonometric functions like sine and tangent have specific signs depending on the quadrant of the angle. Since 180° < α < 270°, α lies in the third quadrant where sine is negative and tangent is positive. Understanding these sign rules is essential to determine the correct value of sin(α/2).
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Quadratic Formula
Half-Angle Formulas
Half-angle formulas allow calculation of trigonometric values for half of a given angle. For sine, the formula is sin(α/2) = ±√((1 - cos α)/2). Choosing the correct sign depends on the quadrant of α/2, which must be determined from the original angle α.
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Relationship Between Tangent and Cosine
Given tan α, you can find cos α using the identity tan α = sin α / cos α and the Pythagorean identity sin² α + cos² α = 1. This step is crucial because the half-angle formula for sine requires the value of cos α.
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