Problem 63 - Precalculus with Limits

Question

In Exercises 61 - 70, prove the identity. \( \sin\left(\dfrac{\pi}{6} + x\right) = \dfrac{1}{2}\left(\cos x + \sqrt{3} \sin x\right) \)

Step-by-Step Solution

Answer

The given identity has been shown to be true by applying the trigonometric identity for the sine of a sum and then substituting the known values of sine and cosine of \( \pi/6 \).

Step 1: Apply the sum of angles identity

Rewrite \( \sin\left(\dfrac{\pi}{6} + x\right) \) as \( \sin\left(\dfrac{\pi}{6}\right)\cos x + \cos\left(\dfrac{\pi}{6}\right)\sin x \). This is done using the trigonometric identity for the sine of a sum, which states that \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)

Step 2: Replace the trigonometric constants

Replace \( \sin\left(\dfrac{\pi}{6}\right) \) with \( \dfrac{1}{2} \) and \( \cos\left(\dfrac{\pi}{6}\right) \) with \( \dfrac{\sqrt{3}}{2} \). These replacements are based on the known values of sine and cosine of \( \pi/6 \)

Step 3: Simplify the expression

Simplify to get \( \dfrac{1}{2}\cos x + \dfrac{\sqrt{3}}{2}\sin x \), which matches the right-hand side of the given identity.

Key Concepts

Sum of Angles Identity

The sum of angles identity is a fundamental tool in trigonometry, allowing us to find the sine, cosine, or tangent of a sum or difference of two angles. Specifically, for sine, the identity can be expressed as:

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

This identity is very handy when you are dealing with expressions involving the addition of angles.
Consider the task of proving identities, like the one given in the exercise where you need to transform \( \sin\left(\frac{\pi}{6} + x\right) \). By using the sum of angles identity, this expression can be rewritten in terms of \( \sin \) and \( \cos \) of individual angles, making it much easier to work with and simplify.
Recognizing that, for specific angle values like \( \frac{\pi}{6} \), we can substitute their known sine and cosine values, thus making calculations straightforward and solving problems efficiently.

Sine Function

The sine function is one of the core trigonometric functions in mathematics. It is defined in the context of a right-angled triangle but is also commonly extended to the unit circle. The sine of an angle \(\theta\) is the y-coordinate of the point on the unit circle at an angle \(\theta\) from the positive x-axis.

• For example, \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\) is derived from the unit circle or by using a 30-60-90 triangle.

Understanding these key facts will help in manipulating and simplifying expressions involving sine, as seen in our exercise.
The sine function is periodic and symmetric, having a period of \(2\pi\). This attribute allows the function to model repeating phenomena, such as sound waves or the seasonal daylight cycle.
Moreover, it has an amplitude of 1, meaning that the sine values will always range from -1 to 1 which is critical when solving and proving identities involving sine.

Cosine Function

The cosine function, like sine, is a foundational component of trigonometry. When you consider a right triangle, the cosine of an angle \(\theta\) is defined as the ratio of the adjacent side to the hypotenuse. Also, much like the sine function, the cosine functions can be represented on the unit circle, where it corresponds to the x-coordinate at a given angle.

• For instance, \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\), which is determined using properties from geometry or directly from the unit circle.

In problems involving both the sine and cosine functions, like our exercise, these values are integral to apply accurately and simplify the identity proofs.
The cosine function is an even function, meaning that it is symmetric about the y-axis. This implies \(\cos(-\theta) = \cos(\theta)\), and it shares the period of \(2\pi\) with sine.
The smooth wave-like graph of the cosine function is useful for modeling cyclical patterns, similar to the sine function, but beginning at its maximum value when \(\theta = 0\).