Textbook Question
In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x
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6. Trigonometric Identities and More Equations
Introduction to Trigonometric Identities
3:41 minutesProblem 3.3.47
Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function. sin(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 \( \sin \left( \frac{\theta}{2} \right) \), which is the sine of half the angle \( \theta \).
Recall the half-angle identity for sine:
\[ \\sin \\left( \\frac{\\theta}{2} \\right) = \\pm \\sqrt{ \\frac{1 - \\cos \\theta}{2} } \]
This formula allows you to find the sine of half an angle if you know \( \cos \theta \).
Determine the sign (positive or negative) of \( \\sin \\left( \\frac{\\theta}{2} \\right) \) based on the quadrant where \( \\frac{\\theta}{2} \) lies. Remember that sine is positive in the first and second quadrants and negative in the third and fourth quadrants.
Find or calculate \( \\cos \\theta \) from the given figure or information. This might involve using the Pythagorean theorem if the figure provides side lengths, or using other trigonometric relationships.
Substitute the value of \( \\cos \\theta \) into the half-angle formula and simplify the expression under the square root. This will give you the exact value of \( \\sin \\left( \\frac{\\theta}{2} \\right) \) up to the sign determined earlier.
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions like sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. Understanding how to identify and calculate these ratios is essential for finding exact values of these functions for a given angle.
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Unit Circle and Angle Measures
The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles. Knowing how to interpret angles on the unit circle helps in finding exact trigonometric values without a calculator.
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Exact Values of Common Angles
Certain angles like 0°, 30°, 45°, 60°, and 90° have well-known exact trigonometric values. Memorizing or deriving these values allows for precise calculation of trigonometric functions without approximation.
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