Textbook Question
In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 6 sin⁴ x
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
3:56 minutesProblem 3.3.49
Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. tan(θ/2)
Verified step by step guidance1
Identify the given angle \( \theta \) and the trigonometric function you need to find. In this case, it appears you need to find \( \tan \left( \frac{\theta}{2} \right) \), the tangent of half the angle \( \theta \).
Recall the half-angle identity for tangent:
\[ \tan \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \]
or alternatively,
\[ \tan \left( \frac{\theta}{2} \right) = \frac{\sin \theta}{1 + \cos \theta} \quad \text{or} \quad \tan \left( \frac{\theta}{2} \right) = \frac{1 - \cos \theta}{\sin \theta} \]
Choose the form that best fits the information given in the figure.
Determine the values of \( \sin \theta \) and \( \cos \theta \) from the figure or from the problem data. This might involve using the coordinates of a point on the unit circle, or lengths of sides in a right triangle related to \( \theta \).
Substitute the values of \( \sin \theta \) and \( \cos \theta \) into the chosen half-angle formula for \( \tan \left( \frac{\theta}{2} \right) \).
Simplify the expression carefully, paying attention to the sign of the tangent function in the quadrant where \( \frac{\theta}{2} \) lies, to find the exact value of \( \tan \left( \frac{\theta}{2} \right) \).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of the Tangent Function
The tangent of an angle θ in a right triangle is the ratio of the length of the side opposite θ to the length of the side adjacent to θ. It can also be expressed as tan(θ) = sin(θ)/cos(θ), linking it to the sine and cosine functions.
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Using Figures to Determine Side Lengths
To find the exact value of a trigonometric function from a figure, identify the lengths of the relevant sides of the triangle. These lengths allow you to compute ratios like tangent accurately, often using known values or the Pythagorean theorem.
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Exact Values of Trigonometric Functions
Exact values refer to precise ratios expressed in simplest radical form or fractions, not decimal approximations. Common angles like 30°, 45°, and 60° have well-known exact values for tangent and other trig functions, which are essential for precise calculations.
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