Problem 55 - Precalculus

Question

For the following exercises, rewrite in terms of \(\sin x\) and \(\cos x .\) $$ \cos \left(x+\frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

\( -\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x \)

Step 1: Identify the Trigonometric Identity

To rewrite \( \cos\left(x + \frac{2\pi}{3}\right) \) in terms of \( \sin x \) and \( \cos x \), we use the cosine addition formula: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

Step 2: Substitute the Values into the Identity

Substitute \( a = x \) and \( b = \frac{2\pi}{3} \) into the identity: \[ \cos\left(x + \frac{2\pi}{3}\right) = \cos x \cos\frac{2\pi}{3} - \sin x \sin\frac{2\pi}{3}. \]

Step 3: Calculate the Exact Values for the Angle

Find \( \cos\frac{2\pi}{3} \) and \( \sin\frac{2\pi}{3} \). Since \( \frac{2\pi}{3} \) is in the second quadrant,\[ \cos\frac{2\pi}{3} = -\frac{1}{2}, \quad \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}. \]

Step 4: Substitute Exact Values and Simplify

Replacing the values in the identity, we get: \[ \cos\left(x + \frac{2\pi}{3}\right) = \cos x \left(-\frac{1}{2}\right) - \sin x \left(\frac{\sqrt{3}}{2}\right). \]Simplifying, we have: \[ \cos\left(x + \frac{2\pi}{3}\right) = -\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x. \]

Key Concepts

Understanding the Cosine Addition Formula

When dealing with trigonometric identities, the cosine addition formula is a powerful tool. It can help us understand how the cosine of the sum of two angles can be expressed in terms of sines and cosines of individual angles. This formula is expressed as: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).
Think of it as a way to "split" the cosine of a sum into more manageable sines and cosines. This becomes particularly useful when dealing with angles that are not common or easy to visualize directly.
For example, in the problem \( \cos(x + \frac{2\pi}{3}) \), each part \( \cos x \) and \( \cos\frac{2\pi}{3} \) can be broken down individually, making the expression simpler to work with. Understanding this formula allows us to easily transform and manipulate trigonometric expressions.

Connecting Sine and Cosine Functions

Sine and cosine functions form the fundamental building blocks of trigonometry. These functions have specific properties:

• Periodic Nature: Both functions repeat their values over regular intervals of \( 2\pi \).
• Interconnected: At any given angle, \( \sin \theta \) and \( \cos \theta \) can be linked through the identity \( \sin^{2} \theta + \cos^{2} \theta = 1 \).

Understanding these relationships is key when expressing one function in terms of another—as seen when rewriting \( \cos(x + \frac{2\pi}{3}) \) using \( \sin x \) and \( \cos x \). This process involves recognizing angles where one or both functions have straightforward values, like the angles \( \frac{2\pi}{3} \), where \( \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \cos\frac{2\pi}{3} = -\frac{1}{2} \).
These remarkable relationships help simplify the comprehension and solving of trigonometric expressions.

Exploring Angle Transformations

Angle transformations in trigonometry involve changing angles through additions or subtractions to make them easier to work with. This often means shifting angles into different quadrants of the unit circle, thereby altering the signs and values of sine and cosine functions associated with those angles.
For instance, when you see the angle \( \frac{2\pi}{3} \), it resides in the second quadrant. Here, cosine values are negative, and sine values remain positive.
Transforming angles like \( x + \frac{2\pi}{3} \) allows for such expressions to be reinterpreted using known values. This transformation is crucial in applying identities like the cosine addition formula, letting us express complex angles as combinations of the primary angle functions, \( \sin\) and \( \cos \), which both vary predictably with transformations across quadrants.
Having a good grasp of how these transformations work provides clarity in handling trigonometric equations and identities efficiently.