Textbook Question
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
cos (x − π/2) = sin x
0. Functions
Trigonometric Identities
7:16 minutesProblem 1.3.38
Textbook Question
What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?
Verified step by step guidance1
Consider the addition formulas for sine and cosine: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \) and \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).
Substitute \( B = 2\pi \) into these formulas. Recall that \( \cos(2\pi) = 1 \) and \( \sin(2\pi) = 0 \).
For the sine addition formula, substitute \( B = 2\pi \): \( \sin(A + 2\pi) = \sin A \cdot 1 + \cos A \cdot 0 = \sin A \).
For the cosine addition formula, substitute \( B = 2\pi \): \( \cos(A + 2\pi) = \cos A \cdot 1 - \sin A \cdot 0 = \cos A \).
These results show that \( \sin(A + 2\pi) = \sin A \) and \( \cos(A + 2\pi) = \cos A \), which agree with the periodicity of sine and cosine functions, confirming that they repeat every \( 2\pi \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Addition Formulas
Addition formulas in trigonometry express the sine and cosine of the sum or difference of two angles. For example, the sine addition formula states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). These formulas are essential for simplifying expressions involving trigonometric functions and for solving equations that include angles.
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Periodic Functions
Trigonometric functions such as sine and cosine are periodic, meaning they repeat their values in regular intervals. The period of sine and cosine is 2π, which indicates that sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x). Understanding periodicity is crucial when analyzing the behavior of trigonometric functions over different intervals.
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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. Using the unit circle helps visualize how angles relate to their sine and cosine values, especially when considering angles like B = 2π.
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