Problem 36 - Precalculus with Limits

Question

In Exercises 29 - 36, write the expression as the sine, cosine,or tangent of an angle. \( \cos 3x \cos 2y + \sin 3x \sin 2y \)

Step-by-Step Solution

Answer

The expression \( \cos 3x \cos 2y + \sin 3x \sin 2y \) can be written as \( \cos (3x - 2y) \)

Step 1: Identify formula

Identify that the given expression \( \cos 3x \cos 2y + \sin 3x \sin 2y \) resembles the formula for the cosine of the difference of two angles i.e., \( \cos (a - b) = \cos a \cos b + \sin a \sin b \). Here \( a = 3x \) and \( b = 2y \)

Step 2: Apply formula

Applying the formula, we rewrite the expression \( \cos 3x \cos 2y + \sin 3x \sin 2y \) as \( \cos (3x - 2y) \)

Key Concepts

Cosine of Difference

The formula for the cosine of the difference of two angles is a fundamental identity in trigonometry. It allows transforming expressions involving cosine and sine into a simpler form. The identity states that:

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

Here, \( a \) and \( b \) are angles in degrees or radians.
The cosine of difference identity is particularly useful because it helps in simplifying expressions where two angles are involved. In the context of this exercise, recognizing that the expression \( \cos 3x \cos 2y + \sin 3x \sin 2y \) matches this pattern helps us directly simplify it to \( \cos (3x - 2y) \).
Understanding this identity not only aids in solving problems but also provides a deeper insight into the interplay between angles in trigonometric functions. This identity and its counterparts, such as sine and tangent of sum and difference, are critical in various applications like solving trigonometric equations and transforming complex expressions.

Sine

The sine function is one of the primary building blocks of trigonometry. It is defined for an angle \( \theta \) as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angle triangle.

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

Its value ranges from -1 to 1, mapping a circular path for an angle moving around a unit circle.
Sine is often used together with cosine in many trigonometric identities, such as the cosine of difference. Its symmetry property is also a key characteristic, where \( \sin(180^\circ - \theta) = \sin \theta \). In the identity \( \cos(a - b) = \cos a \cos b + \sin a \sin b \), sine helps in balancing the equation when angles are subtracted.
Additionally, mastering the sine function and its properties enhances one's ability to comprehend wave patterns and oscillations, providing a broad application in fields like physics and engineering.

Cosine

Cosine is another pillar of trigonometric functions, often paired with sine to form foundational identities. It represents the ratio of the side adjacent to the angle to the hypotenuse in a right-angled triangle:

• \( \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

This function is crucial for its periodic properties and its role in determining angles and distances in geometric contexts.
The cosine function also has values ranging from -1 to 1, and interestingly, \( \cos(\theta) = \sin(90^\circ - \theta) \), showing its complementary nature to sine. It's instrumental in the formula \( \cos(a - b) = \cos a \cos b + \sin a \sin b \), showcasing how the cosine of one angle and the sine of another can combine through addition and subtraction to yield these trigonometric identities.
A strong grasp of the cosine function can significantly empower a student's understanding of wave forms, which are prevalent in sound engineering, electrical engineering, and other applied sciences.