Textbook Question
In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x
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6. Trigonometric Identities and More Equations
Solving Trigonometric Equations Using Identities
5:11 minutesProblem 3.3.55c
Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:
c. tan(α/2)
tan α = 4/3, 180° < α < 270°
Verified step by step guidance1
Identify the given information: \( \tan \alpha = \frac{4}{3} \) and the angle \( \alpha \) lies in the third quadrant where \( 180^\circ < \alpha < 270^\circ \).
Recall that in the third quadrant, both sine and cosine values are negative, but tangent is positive, which is consistent with the given \( \tan \alpha = \frac{4}{3} \).
Use the identity for tangent of half an angle: \( \tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} \) or use the formula \( \tan \frac{\alpha}{2} = \frac{\sin \alpha}{1 + \cos \alpha} \) or \( \tan \frac{\alpha}{2} = \frac{1 - \cos \alpha}{\sin \alpha} \). Choose the form that is easiest based on known values.
Find \( \sin \alpha \) and \( \cos \alpha \) using the given \( \tan \alpha = \frac{4}{3} \). Since \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \), set \( \sin \alpha = -4k \) and \( \cos \alpha = -3k \) (both negative in the third quadrant), then use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to solve for \( k \).
Substitute the values of \( \sin \alpha \) and \( \cos \alpha \) into the half-angle formula chosen in step 3 to find \( \tan \frac{\alpha}{2} \). Determine the correct sign of \( \tan \frac{\alpha}{2} \) based on the quadrant where \( \frac{\alpha}{2} \) lies.
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Function and Its Properties
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. It can also be defined as sin(α)/cos(α) on the unit circle. Understanding how tangent behaves in different quadrants is essential for determining the sign and value of tan(α).
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Introduction to Tangent Graph
Reference Angles and Quadrant Location
The given angle α lies between 180° and 270°, which places it in the third quadrant. In this quadrant, both sine and cosine are negative, but tangent is positive. Using the reference angle helps find exact trigonometric values by relating α to a known acute angle.
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Half-Angle Formulas
Half-angle formulas allow calculation of trigonometric functions of half an angle using the functions of the original angle. For tangent, the half-angle formula expresses tan(α/2) in terms of sin(α) and cos(α), enabling the exact value of tan(α/2) to be found from tan(α).
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