Problem 91 - Precalculus with Limits

Question

In Exercises 91 - 94, determine whether the statement is true or false. Justify your answer. \( \sin(u \pm v) = \sin u \cos v \pm \cos u \sin v \)

Step-by-Step Solution

Answer

The given statement is true.

Step 1: Identify the identity

It is known in trigonometry that the sine of the sum/difference of two angles can be expressed as follows: \( \sin(u + v) = \sin u \cos v + \cos u \sin v \) and \( \sin(u - v) = \sin u \cos v - \cos u \sin v \). These are the angle sum and difference identities for sine.

Step 2: Compare with the given identity

Upon comparing the given identity \( \sin(u \pm v) = \sin u \cos v \pm \cos u \sin v \) with the known identities, it can be seen that they match the known identities.

Step 3: Confirm the truth of the identity

Since the given identity matches perfectly with the known identities, it can be concluded that the provided statement is true.

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions, essential in both geometry and calculus. It takes an angle as its input and provides the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. In mathematical terms, for a given angle \( \theta \), the sine function is denoted as \( \sin(\theta) \).

• Its value ranges between -1 and 1.
• It is periodic, repeating every \( 2\pi \) radians or 360 degrees.
• It is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \).

The sine function is often encountered in wave motion, oscillations, and more. Understanding it is crucial for diving deeper into trigonometry.

Angle Sum and Difference

In trigonometry, angle sum and difference identities are powerful tools for simplifying expressions involving angles. These identities allow us to express trigonometric functions of the sum or difference of angles in terms of the functions of the individual angles.
For sine, the angle sum and difference identities are given by:

• \( \sin(u + v) = \sin u \cos v + \cos u \sin v \)
• \( \sin(u - v) = \sin u \cos v - \cos u \sin v \)

These identities can be particularly helpful when solving equations or proving other trigonometric identities. They depict how the sine function behaves when the angle is expressed as a composition of two smaller angles.

Trigonometric Functions

Trigonometric functions extend beyond just sine to include cosine, tangent, and their reciprocals. These functions are fundamental in mathematics, enabling the analysis of triangles, modeling periodic phenomena, and providing solutions in calculus and physics.
Key trigonometric functions include:

• Sine \( (\sin) \): ratio of the opposite side to the hypotenuse.
• Cosine \( (\cos) \): ratio of the adjacent side to the hypotenuse.
• Tangent \( (\tan) \): ratio of the opposite side to the adjacent side.

Trigonometric functions are cyclic, with sine and cosine having a period of \( 2\pi \), while tangent has a period of \( \pi \). These functions are used in various fields such as engineering, physics, and computer science for analyzing waves, signals, and more.