Textbook Question
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (A − B) = sin A cos B − cos A sin B
0. Functions
Trigonometric Identities
5:36 minutesProblem 1.3.46
Textbook Question
Evaluate sin (5π/12).
Verified step by step guidance1
Recognize that \( \frac{5\pi}{12} \) is not a standard angle for which we know the sine value directly. We can express it as a sum or difference of angles for which we know the sine and cosine values.
Express \( \frac{5\pi}{12} \) as a sum of two angles: \( \frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} \). Both \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) are standard angles.
Use the sine addition formula: \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \). Here, \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \).
Substitute the known values: \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \), \( \cos(\frac{\pi}{3}) = \frac{1}{2} \), \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), and \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \).
Calculate \( \sin(\frac{5\pi}{12}) = \sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) + \cos(\frac{\pi}{3})\sin(\frac{\pi}{4}) \) by substituting the values and simplifying the expression.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate angles to ratios of sides in right triangles. The sine function specifically measures the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding these functions is essential for evaluating angles and solving problems in trigonometry and calculus.
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Angle Addition Formulas
The angle addition formulas allow us to express the sine of a sum of angles in terms of the sines and cosines of the individual angles. For example, sin(a + b) = sin(a)cos(b) + cos(a)sin(b). This concept is particularly useful for evaluating sine at angles that are not standard, such as 5π/12, by breaking it down into known angles.
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Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric interpretation of trigonometric functions, where the x-coordinate represents the cosine and the y-coordinate represents the sine of an angle. Understanding the unit circle is crucial for evaluating trigonometric functions at various angles, including those expressed in radians.
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