Problem 123 - Precalculus with Limits

Question

In Exercises 111 - 124, verify the identity. \( \sin \left(\dfrac{\pi}{6} + x\right) + \sin\left(\dfrac{\pi}{6} - x\right) = \cos x \)

Step-by-Step Solution

Answer

The given identity is verified as the two expressions on either side of the equation are equal. Thus, \(\sin \left(\dfrac{\pi}{6} + x\right) + \sin\left(\dfrac{\pi}{6} - x\right) = \cos x\) is a true identity.

Step 1: Break down the expressions

Start by using the sum-to-product identity for sine, which is \(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\). Apply this to both the \(\sin(\dfrac{\pi}{6} + x)\) and the \(\sin(\dfrac{\pi}{6} - x)\) expressions.

Step 2: Continue simplification

After applying the formula, we have: \(\sin \frac{\pi}{6} \cos x + \cos \frac{\pi}{6} \sin x + \sin \frac{\pi}{6} \cos x - \cos \frac{\pi}{6} \sin x\). This simplifies to \(2 \sin \frac{\pi}{6} \cos x.\)

Step 3: Use the known value of sine

The value of \(\sin \frac{\pi}{6}\) is \(\frac{1}{2}\); thus, \(2\sin \frac{\pi}{6} \cos x = \cos x\).

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are essential tools in trigonometry that allow us to transform sums or differences of trigonometric functions into products. This can significantly simplify complicated expressions and is particularly useful when verifying identities or solving equations.
In the example at hand, we rely on the sum-to-product identity for sine:

• \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)

This formula helps us to express a sine addition or subtraction as a combination of sine and cosine terms. When applied to the exercises given, this identity allows simplification and consolidation of similar components, ultimately reducing them to a more manageable form.
Understanding these identities not only enriches your trigonometric toolkit but also enhances problem-solving skills by providing a structured way to break down and simplify complex expressions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in the study of triangles, especially when dealing with right-angled ones. They provide relationships between the angles and sides of a triangle and extend to describing periodic phenomena.
In our example, the sine function is primarily used. The sine function \( \sin \theta \) represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. Similarly, the cosine function \( \cos \theta \) represents the ratio of the adjacent side to the hypotenuse.
These functions are not only applicable for real-life measurements in terms of distance and angles but also critical in various fields such as physics, engineering, and computer science. They set the foundation for understanding oscillatory motion, wave dynamics, and signal processing.
Mastering trigonometric functions opens a wide door of applications and gives you a robust method for analyzing and predicting periodic behavior in both natural and synthetic processes.

Sine and Cosine

The sine and cosine functions are two of the most vital trigonometric functions used extensively in math and science to model oscillations and waves. They are periodic functions, meaning they repeat values at regular intervals. For sine and cosine, this interval is \(2\pi\) radians or 360 degrees.

• The sine function, \( \sin x \), starts at zero and reaches its peak of 1 at 90 degrees before returning to zero at 180 degrees, going to -1 at 270 degrees, and finally completing the cycle at 360 degrees.
• The cosine function, \( \cos x \), starts at 1 and follows a similar pattern but is displaced by 90 degrees compared to sine. Thus \( \cos x = \sin(x + \pi/2) \).

These functions are incredibly useful for modeling cyclical phenomena, such as alternating current in electricity or sound waves in acoustics.
In the context of the given trigonometric identity exercise, realizing the specific values of sine and cosine at key angles (like \( \frac{\pi}{6} \)) was crucial for simplifying the expression. Remembering the sine for \( \frac{\pi}{6} \) is \( \frac{1}{2} \) and cosine for \( \frac{\pi}{6} \) is \( \frac{\sqrt{3}}{2} \) helps resolve many trigonometric problems efficiently.