Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function.θtan -------2
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
5:11 minutesProblem 55
Textbook Question
In Exercises 55–58, use the given information to find the exact value of each of the following:αc. tan ------24tan α = ------ , 180° < α < 270°3
Verified step by step guidance1
Recognize that the problem involves finding \( \tan \left( \frac{\alpha}{2} \right) \) given \( \tan \alpha = \frac{4}{3} \) and \( 180^\circ < \alpha < 270^\circ \). This means \( \alpha \) is in the third quadrant.
Recall the half-angle identity for tangent: \( \tan \left( \frac{\alpha}{2} \right) = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} \).
Since \( \alpha \) is in the third quadrant, both sine and cosine are negative. Use the identity \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \) to express \( \sin \alpha \) and \( \cos \alpha \) in terms of \( \tan \alpha \).
Use the Pythagorean identity \( \sin^2 \alpha + \cos^2 \alpha = 1 \) to find \( \sin \alpha \) and \( \cos \alpha \). Substitute \( \tan \alpha = \frac{4}{3} \) into \( \sin \alpha = \frac{4}{5} \) and \( \cos \alpha = -\frac{3}{5} \) (since both are negative in the third quadrant).
Substitute \( \cos \alpha = -\frac{3}{5} \) into the half-angle identity to find \( \tan \left( \frac{\alpha}{2} \right) \). Since \( \frac{\alpha}{2} \) is in the second quadrant, choose the positive value for \( \tan \left( \frac{\alpha}{2} \right) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Function
The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ)/cos(θ). Understanding the properties of the tangent function, including its periodicity and behavior in different quadrants, is essential for solving trigonometric equations.
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Quadrants of the Unit Circle
The unit circle is divided into four quadrants, each corresponding to specific ranges of angle measures. In the third quadrant (180° < α < 270°), both sine and cosine values are negative, which affects the sign of the tangent function. Recognizing the quadrant in which an angle lies helps determine the sign of trigonometric values and is crucial for finding exact values.
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Exact Values of Trigonometric Functions
Exact values of trigonometric functions can be derived from special angles (like 30°, 45°, and 60°) or by using known values and identities. For example, if tan(α) is given as a fraction, one can use the properties of triangles or the unit circle to find the exact value of tan(α/2) using the half-angle formula. Mastery of these techniques is vital for accurately solving trigonometric problems.
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