Problem 35 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \cos \left(x-\frac{\pi}{2}\right)=\sin x $$

Step-by-Step Solution

Answer

The result of \(\cos \left(x-\frac{\pi}{2}\right) = \sin x\) after applying difference identities proves that the provided function is indeed correct.

Step 1: Recall Sum and Difference Identities

Recall the sum and difference identities for cosine: \[\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \]

Step 2: Apply the Identity to the Problem

Apply the identity to the given problem, with \(a = x\) and \(b = \frac{\pi}{2}\):\[\cos \left(x-\frac{\pi}{2}\right) = \cos(x)\cos \left(\frac{\pi}{2}\right) + \sin(x)\sin \left(\frac{\pi}{2}\right)\]

Step 3: Evaluate Cosine and Sine Values

Evaluate the cosine of \(\frac{\pi}{2}\) and the sine of \(\frac{\pi}{2}\): \[\cos \left(x-\frac{\pi}{2}\right) = \cos(x) * 0 + \sin(x) * 1 = \sin(x)\]

Key Concepts

Sum and Difference Identities

Sum and difference identities are a very important tool in trigonometry that help simplify expressions and solve equations involving trigonometric functions. These identities express trigonometric functions of sums or differences of angles in terms of functions of individual angles. Here's what you need to know:

• The sum identity for cosine: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
• The difference identity for cosine: \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \)
• The sum identity for sine: \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
• The difference identity for sine: \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \)

Using these identities, we can expand and simplify trigonometric expressions, helping in verifying or proving equality between different terms. The application in exercises often involves plugging in specific angle values and evaluating the resulting expressions.

Cosine and Sine Functions

The cosine and sine functions are fundamental in the study of trigonometry. They are one of the basic building blocks for understanding angular relationships in mathematical and real-world applications.

• Cosine function: The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. Its value ranges from -1 to 1. For example, \( \cos(0) = 1 \) and \( \cos\left(\frac{\pi}{2}\right) = 0 \).
• Sine function: The sine of an angle also ranges from -1 to 1 and is defined as the ratio of the length of the opposite side to the hypotenuse. For instance, \( \sin(0) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \).

These functions are periodic, which means they repeat their values in regular intervals. The cosine function is an even function, meaning \( \cos(-x) = \cos(x) \), while the sine function is an odd function, indicating \( \sin(-x) = -\sin(x) \). This symmetry property is crucial in various proofs and problem-solving approaches in trigonometry.

Verifying Trigonometric Identities

Verifying trigonometric identities involves showing that two trigonometric expressions are equivalent. This process often requires manipulating one or both sides of the equation until they match.Here are some tips to effectively verify identities:

• Start by applying known identities, such as sum/difference identities, to express terms in another form.
• Simplify expressions using known values of trigonometric functions, such as \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \).
• Consider transforming both sides of the equation into simpler or alternative forms, which might reveal equivalence.

In the given exercise, you used the difference identity for cosine, substituting \(a = x\) and \(b = \frac{\pi}{2}\), to verify that \( \cos \left(x-\frac{\pi}{2}\right) = \sin x \). The process of verification showcases how strategic simplifications, along with correct applications of identities, lead to proving an identity effectively.