Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the expression as the sine, cosine, or tangent of an angle. $$\sin 3.5 x \cos 1.2 y+\cos 3.5 x \sin 1.2 y$$

Step-by-Step Solution

Answer

The given expression \( \sin 3.5x \cos 1.2y + \cos 3.5x \sin 1.2y \) is equivalent to \( \sin(3.5x + 1.2y) \).

Step 1: Identification of the Identity

Identify the trigonometric identity in the given problem. In this case, the expression \( \sin 3.5x \cos 1.2y+ \cos 3.5x \sin 1.2y \) applies to the formula for the sine of two angles summed together, which is \( \sin(A + B) = \sin A \cos B + \cos A \sin B \). Here, A is equivalent to 3.5x, and B is equivalent to 1.2y.

Step 2: Applying the Trigonometric Identity

Apply the trigonometric identity to the given expression. According to the formula, it's clear that the given expression equals \( \sin(A + B) \). Replacing A with 3.5x and B with 1.2y, the equivalent expression is \( \sin(3.5x + 1.2y) \).

Key Concepts

Sine and Its Role in Trigonometry

In trigonometry, the sine function is a fundamental concept that relates an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is denoted as \( \sin \theta \), where \( \theta \) is the angle.

• Sine is often used to solve problems involving triangles, waves, and oscillations.
• It oscillates between -1 and 1, and has a periodicity of \(2\pi\).

These properties make the sine function a versatile and powerful tool in both mathematical and practical applications. In the given exercise, the expression \(\sin 3.5x\) is part of a sum-of-angles formula. This highlights how sine can be combined with other trigonometric functions to simplify complex expressions.

Understanding Cosine

The cosine function is another crucial trigonometric function. It measures the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. Cosine is denoted by \( \cos \theta \), similar to sine, with \( \theta \) representing the angle.

• The cosine function is also periodic, with the same period as sine, \(2\pi\).
• It ranges from -1 to 1, taking values at specific angles, crucial in calculations.

In various contexts, like the given exercise, cosine often interacts with the sine function. Here, \(\cos 1.2y\) appears in a combination that leverages these interactions to use a known identity, simplifying expressions.

Sum of Angles Formula

A fascinating aspect of trigonometry is how we can combine angles using specific identities known as sum of angles formulas. The sum of angles for sine is \[ \sin(A + B) = \sin A \cos B + \cos A \sin B \].This identity allows us to express the sine of a sum of two angles in terms of sines and cosines of the individual angles.

• Understanding this formula is essential for simplifying complex trigonometric expressions.
• It is particularly useful in solving problems where angles can be combined or decomposed into simpler parts.

In the given exercise, the sum of angles identity reveals that \( \sin 3.5x \cos 1.2y + \cos 3.5x \sin 1.2y \) simplifies to \( \sin(3.5x + 1.2y) \). This simplification is a powerful application of trigonometric identities, making calculations more manageable.