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

\( \cos(\pi + x) = -\cos x \).

Step 1: Use the Trigonometric Identity

Recall the trigonometric identity for cosine of a sum: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Here, we will use this identity to express \( \cos(\pi + x) \).

Step 2: Apply the Identity

Using the identity from Step 1, set \( a = \pi \) and \( b = x \). Then, \( \cos(\pi + x) = \cos \pi \cos x - \sin \pi \sin x \).

Step 3: Substitute Known Values

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

Step 4: Simplify the Expression

Simplify the equation from Step 3: \( \cos(\pi + x) = -\cos x \).

Key Concepts

Cosine Addition Formula

The cosine addition formula is a fundamental trigonometric identity used to find the cosine of a sum of two angles. It is expressed as:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)

This formula allows us to break down complex trigonometric expressions into simpler parts based on known values of cosine and sine.
It is particularly helpful for expressing compound angles in terms of basic trigonometric functions. When faced with expressions like \( \cos(\pi + x) \), the addition formula helps us express it as the product of sines and cosines of individual angles.
By systematically applying this formula, we can transform complex trigonometric expressions into more familiar terms. Understanding how to utilize the cosine addition formula is key to solving a variety of trigonometric problems.

Cosine Transformation

Cosine transformations involve altering the angle in a cosine function to produce different expressions. In the context of the given exercise, the angle \( \pi + x \) undergoes transformation using the cosine addition formula. The transformation helps simplify the expression, allowing for easier manipulation and understanding.After substituting the values into the formula, you get:

• \( \cos(\pi + x) = \cos \pi \cos x - \sin \pi \sin x \)

Then, rely on known trigonometric constants where \( \cos \pi = -1 \) and \( \sin \pi = 0 \) to simplify this further:

• \( \cos(\pi + x) = -\cos x - 0 \)
• The resulting expression is \( \cos(\pi + x) = -\cos x \)

This step demonstrates how transformations can help in simplifying trigonometric expressions by using basic identities and values.

Trigonometric Values

Trigonometric values refer to the specific sine and cosine values of common angles. Knowing these values is crucial as they are often used in simplifying expressions and solving problems.For example:

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

These values are based on the unit circle, which is a powerful tool in trigonometry.
The unit circle provides all common angle values for cosine and sine and is essential for solving trigonometric equations.
Using these values in conjunction with trigonometric identities like the cosine addition formula, we can express complex angles, such as \( \pi + x \), in terms of simpler expressions involving \( \sin x \) and \( \cos x \). Remember, a solid grasp of these fundamental values will vastly improve your ability to handle trigonometric calculations with ease.