Problem 39 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

\(-\cos x\)

Step 1: Recall the Identity for Cosine of Sum

We begin by using the identity for the cosine of a sum: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Here, \(a = \pi\) and \(b = x\).

Step 2: Substitute Into the Identity

Substitute \(a = \pi\) and \(b = x\) into the identity: \(\cos(\pi + x) = \cos \pi \cos x - \sin \pi \sin x\).

Step 3: Evaluate Trigonometric Functions of \(\pi\)

We know from the unit circle that \(\cos \pi = -1\) and \(\sin \pi = 0\). Substitute these values into the equation: \(\cos(\pi + x) = (-1)\cos x - (0)\sin x\).

Step 4: Simplify the Expression

Simplify the expression: \(\cos(\pi + x) = -\cos x\). Thus, the quantity is expressed in terms of \(\cos x\).

Key Concepts

Cosine of Sum Formula

The Cosine of Sum Formula is a handy identity in trigonometry. It helps to simplify expressions where you have a sum inside the cosine function. The formula is expressed as \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). In simpler terms, it breaks down the cosine of a sum into a combination of sines and cosines of the individual angles.
Learning this identity enables you to handle more complex trigonometric expressions. Whether you're dealing with acute angles or standard angles like \( \pi \), the principle stays the same. Keep in mind:

• Cosine deals with horizontal components.
• Sine deals with vertical components.

By applying this formula, you can transform a seemingly complex problem into a simpler one. It is especially beneficial when combining known angles to find unknown values.

Unit Circle

The unit circle is a fundamental tool in trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate plane. The angle created by a point on the unit circle is measured from the positive x-axis.
Points on the unit circle correspond to coordinates \( (\cos \theta, \sin \theta) \). This pairing is pivotal for memorizing trigonometric functions for key angles like \( \pi \) and \( \frac{\pi}{2} \).
For \( \pi \), the coordinates become \( (-1, 0) \), meaning:

• \( \cos \pi = -1 \)
• \( \sin \pi = 0 \)

By understanding the unit circle, you can easily evaluate trigonometric functions at a glance. Whether working with angles in radians or degrees, the unit circle helps you see the relationships between angles and their sine or cosine values.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and known angle values to reduce the complexity of an equation. This process helps in solving trigonometric equations more efficiently.
If you apply identities like the cosine of sum formula, substitution becomes straightforward. For example:

• Use \( \cos(\pi + x) = \cos \pi \cos x - \sin \pi \sin x \)
• Substitute known values: \( \cos \pi = -1 \) and \( \sin \pi = 0 \)
• Calculate: \( \cos(\pi + x) = - \cos x \)

This process allows converting expressions into more manageable terms in relation to \( \sin x \) and \( \cos x \).
Overall, simplifying ensures understanding and easy manipulation of trigonometric equations during problem-solving. It involves breaking down complex expressions into familiar terms, assisting in deeper comprehension of how trigonometric functions interact.