Problem 52 - Precalculus

Question

For the following exercises, rewrite in terms of \(\sin x\) and \(\cos x .\) $$ \sin \left(x+\frac{11 \pi}{6}\right) $$

Step-by-Step Solution

Answer

\(\sin \left(x+\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x \).

Step 1: Recall the angle addition formula

The angle addition formula for sine is given by: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). We will apply this formula to rewrite \( \sin \left(x+\frac{11\pi}{6}\right) \).

Step 2: Identify the elements in the formula

Identify \( a = x \) and \( b = \frac{11\pi}{6} \). Then, apply these to the angle addition formula: \( \sin \left(x+\frac{11\pi}{6}\right) = \sin x \cos \left(\frac{11\pi}{6}\right) + \cos x \sin \left(\frac{11\pi}{6}\right) \).

Step 3: Evaluate trigonometric functions at \(\frac{11\pi}{6}\)

\( \frac{11\pi}{6} \) is equivalent to \( 2\pi - \frac{\pi}{6} \), positioning it in the fourth quadrant where sine is negative and cosine is positive.- \( \cos \left(\frac{11\pi}{6}\right) = \cos \left(2\pi - \frac{\pi}{6}\right) = \sqrt{3}/2 \).- \( \sin \left(\frac{11\pi}{6}\right) = \sin \left(2\pi - \frac{\pi}{6}\right) = -1/2 \).

Step 4: Substitute the values back into the formula

Substitute the evaluated values back into the angle addition formula: \[\sin \left(x+\frac{11\pi}{6}\right) = \sin x \cdot \frac{\sqrt{3}}{2} + \cos x \cdot \left(-\frac{1}{2}\right) \].This simplifies to: \[\sin \left(x+\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x \].

Key Concepts

Angle Addition Formulas

The angle addition formulas are key tools in trigonometry used to find the sine, cosine, and tangent of sums or differences of angles. These formulas make it possible to break down trigonometric expressions into simpler components, which is especially useful when dealing with non-standard angles. For sine, the angle addition formula is:

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

For cosine, the formula is:

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

Let's use these formulas when angles are not directly accessible with our common reference angles like 0, \( \pi/2 \), \( \pi \), and so on.
In our example, transforming \( \sin \left(x+\frac{11\pi}{6}\right) \) requires using the sine angle addition formula.
By applying this formula, we substituted the components of the angle \( x \) and \( b = \frac{11\pi}{6} \) into \( \sin(a + b) \), ultimately arriving at the expression in terms of \( \sin x \) and \( \cos x \).

Sine and Cosine Functions

The sine and cosine functions are foundational elements in trigonometry representing the y-coordinate and x-coordinate, respectively, of a point on the unit circle corresponding to a given angle. This fundamental relationship is crucial when manipulating trigonometric expressions and understanding their behavior in different quadrants.
The unit circle definition helps us identify the values of sine and cosine for specific angles. For example, when we consider angles like \( \frac{11\pi}{6} \), understanding its positioning on the unit circle allows us to easily determine its sine and cosine values:

• \( \cos \left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2} \)
• \( \sin \left(\frac{11\pi}{6}\right) = -\frac{1}{2} \)

These values are determined by the corresponding y and x coordinates at that angle, using the properties of the unit circle.
Knowing these values is essential, as it provides us with the components needed to plug into our angle addition formulas, assisting in the simplification and rewriting of trigonometric expressions.

Trigonometric Simplification

Trigonometric simplification involves the process of reducing complex trigonometric expressions to simpler forms. This is particularly important when solving equations or manipulating expressions for easier evaluation.
In the provided exercise, simplification was done by applying the angle addition formula and substituting known sine and cosine values of specific angles:

• Tackling \( \sin \left(x+\frac{11\pi}{6}\right) \) using the angle addition formula results in \( \sin x \cos \left(\frac{11\pi}{6}\right) + \cos x \sin \left(\frac{11\pi}{6}\right) \). By inserting known values, it transitions to a more workable form: \( \frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x \).

Simplification like this is vital as it transforms what initially seems complex into a function readily expressible in terms of simpler sine and cosine functions. These transformations often reveal fundamental patterns and relations that allow for further explorations in mathematical problem-solving and analysis. Through practice and familiarity, identifying which identities and simplifications to employ becomes intuitive.