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Chapter 7: Problem 12

Rewrite in terms of \(\sin (x)\) and \(\cos (x)\). \(\cos \left(x+\frac{2 \pi}{3}\right)\)

Short Answer

Expert verified
\( \cos(x+\frac{2\pi}{3}) = -\frac{1}{2} \cos(x) - \frac{\sqrt{3}}{2} \sin(x) \)

Step by step solution

01

Use the Cosine Addition Formula

The cosine addition formula is given by: \( \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \). For the expression \( \cos \left( x + \frac{2\pi}{3} \right) \), set \( a = x \) and \( b = \frac{2\pi}{3} \). Apply the formula to obtain: \( \cos \left( x + \frac{2\pi}{3} \right) = \cos(x) \cos \left( \frac{2\pi}{3} \right) - \sin(x) \sin \left( \frac{2\pi}{3} \right) \).
02

Evaluate Trigonometric Values for Specific Angles

The values of the cosine and sine functions for \( \frac{2\pi}{3} \) are: \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \) and \( \sin \left( \frac{2\pi}{3} \right) = \frac{\sqrt{3}}{2} \). Substitute these values into the expression from the previous step.
03

Substitute and Simplify

Substitute the trigonometric values from Step 2 into the expression: \( \cos \left( x + \frac{2\pi}{3} \right) = \cos(x) \left(-\frac{1}{2}\right) - \sin(x) \left(\frac{\sqrt{3}}{2}\right) \). Simplify this expression to get: \( \cos \left( x + \frac{2\pi}{3} \right) = -\frac{1}{2} \cos(x) - \frac{\sqrt{3}}{2} \sin(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Addition Formula
The cosine addition formula is a key identity in trigonometry that allows us to simplify expressions involving the cosine of sums of angles. By using the formula:
  • \( \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \)
we can expand the cosine of a sum of two angles \(a\) and \(b\) into simpler trigonometric terms. This formula helps in converting complex trigonometric expressions into forms that are easier to evaluate or simplify.
In the exercise, we applied this formula to the expression \( \cos \left( x + \frac{2\pi}{3} \right) \) by setting \( a = x \) and \( b = \frac{2\pi}{3} \). This allowed us to break down the problem into components involving \( \cos(x) \) and \( \sin(x) \), which are fundamental trigonometric functions.
Trigonometric Values
Trigonometric values for specific angles play a crucial role in simplifying expressions. These values are often memorized for common angles such as \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), and so on.
For the angle \( \frac{2\pi}{3} \), the trigonometric values that are particularly important are:
  • \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \)
  • \( \sin \left( \frac{2\pi}{3} \right) = \frac{\sqrt{3}}{2} \)
Using these specific values is essential when substituting back into the cosine addition formula, as it allows the calculation to be completed. Recognizing and using these values correctly is a fundamental part of solving trigonometric problems involving angles.
Angle Conversion
Angle conversion involves transforming angles from one unit or format to another, ensuring they are suitable for calculation. In many trigonometry problems, it is crucial to express angles in radians, since this is the standard measure within the context of trigonometric functions.
When working with angles like \( \frac{2\pi}{3} \), remember:
  • Angles in radians can be directly input into trigonometric functions in calculus and other advanced mathematics.
  • This angle, \( \frac{2\pi}{3} \), is equivalent to \(120^\circ\) in degrees.
Understanding angle conversion ensures that calculations are performed consistently and accurately, particularly when working within the realm of trigonometric identities and equations.

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