Problem 40 - Thomas Calculus

Question

In Exercises \(39-42,\) express the given quantity in terms of \(\sin x\) and \(\cos x .\) $$ \sin (2 \pi-x) $$

Step-by-Step Solution

Answer

\(\sin(2\pi - x) = -\sin(x)\)

Step 1: Understand the Trigonometric Identity

The sine of an angle subtracted from \(2\pi\) is equal to the negative sine of the angle itself. This is based on the identity \(\sin(2\pi - \theta) = -\sin(\theta)\).

Step 2: Apply the Identity to the Given Expression

Apply the identity to the expression \(\sin(2\pi - x)\). According to the identity, we have \(\sin(2\pi - x) = -\sin(x)\).

Step 3: Express in Terms of \(\sin x\) and \(\cos x\)

The expression \(-\sin(x)\) is already in terms of \(\sin x\) since it is simply the negation of \(\sin x\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions and is often symbolized as \( \sin \). It applies primarily within the context of angles and is defined for a right-angled triangle as the ratio of the length of the opposite side to the angle and the hypotenuse. This mathematical tool helps in analyzing wave patterns and oscillatory behaviors.

Some critical aspects of the sine function include:

• It is a periodic function with a period of \( 2\pi \).
• The range of \( \sin \theta \) is \([-1, 1]\).
• The sine function is odd, which means \( \sin(-\theta) = -\sin(\theta) \).

These characteristics make the sine function very useful in physics, engineering, and other fields that involve wave dynamics. Understanding the sine function is pivotal for recognizing how angles and lengths interact geometrically and trigonometrically.

Cosine Function

The cosine function, denoted as \( \cos \), is another essential trigonometric function that is intertwined with the sine function. Like sine, it is defined for a right-angled triangle but reflects the ratio of the length of the adjacent side to the angle over the hypotenuse. This function, too, is crucial in understanding oscillatory behavior.

Key features of the cosine function include:

• It is periodic with a period of \( 2\pi \), similar to the sine function.
• The range of the cosine function is also \([-1, 1]\).
• The cosine function is even, meaning \( \cos(-\theta) = \cos(\theta) \).

These properties make the cosine function equally important in mathematical modeling and scientific applications, particularly when dealing with cycles and rotations.

Angle Subtraction Identities

Angle subtraction identities are an integral part of trigonometry that help simplify expressions involving trigonometric functions. A critical identity within this group is the sine angle subtraction identity, which allows the simplification of the expression \( \sin(2\pi - x) \) into a more manageable form.

The identity \( \sin(2\pi - \theta) = -\sin(\theta) \) tells us that taking the sine of \(2\pi \) minus any angle \( x \) results in the negative sine of the angle \( x \) itself.

• This identity utilizes the periodicity of the sine function, using its symmetrical properties.
• These identities are indispensable in trigonometric simplifications and solving equations involving trigonometric functions.

With the sine and cosine identities, you can transform complex trigonometric expressions into simpler ones, making problem-solving more straightforward and efficient.