Question

Use a graphing utility to find an equivalent expression for $\cos \left(x-\frac{\pi}{2}\right)$.

   Use a graphing utility to find an equivalent expression for $\cos \left(x-\frac{\pi}{2}\right)$.

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 6, Problem 91

Instant Answer

Solved by Expert Joshua Eastwood

04/12/2020

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The cosine function is a shifted version of the sine function. Specifically, $\cos(x) = \sin\left(x + \frac{\pi}{2}\right)$. Show more…

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Use a graphing utility to find an equivalent expression for $\cos \left(x-\frac{\pi}{2}\right)$.

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Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations involving trigonometric functions that allow one to rewrite expressions in different, yet equivalent, forms. In this context, the cosine difference identity, which states that cos(a - b) = cos a cos b + sin a sin b, is key to rewriting cos(x - ?/2) into an alternative expression.

Phase Shifts in Trigonometric Functions

Phase shifts refer to the horizontal translations of trigonometric functions on the graph. Recognizing that subtracting ?/2 from x effectively shifts the cosine function helps in understanding how it relates to other trigonometric functions such as sine, since such shifts can transform one function into another.

Graphing Utilities for Function Verification

Graphing utilities are software tools that allow one to visualize functions and verify their properties. They can be used to confirm that two different algebraic expressions represent the same function by comparing their graphs over a range of values.

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