Problem 34 - Contemporary Precalculus

Question

In Exercises \(31-36\), write the expression as a single real mum. ber. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\cos (2 \pi / 3) \cos (\pi)+\sin (2 \pi / 3) \sin (\pi)$$

Step-by-Step Solution

Answer

Answer: The value of the expression is \(\boxed{\frac{1}{2}}\).

Step 1: Identify the trigonometric identities

We will use the following trigonometric identities to simplify the given expression: 1. \(\cos(\pi) = -1\) 2. \(\sin(\pi) = 0\) 3. Angle addition formula: \(\cos(a+b) = \cos a \cos b - \sin a \sin b\)

Step 2: Calculate the value of the trigonometric functions in the expression

By using the first two trigonometric identities: \(\cos (2 \pi / 3) \cos (\pi) = \cos (2 \pi / 3) (-1)\) \(\sin (2 \pi / 3) \sin (\pi) = \sin (2 \pi / 3) (0)\)

Step 3: Simplify the expression

Our expression becomes: \((-1)\cos (2 \pi / 3)+0\)

Step 4: Find the value of \(\cos (2 \pi / 3)\)

Using the third trigonometric identity: \(\cos (2 \pi / 3) = \cos(\pi/3 + \pi/3)\) Now apply the cosine addition formula, \(\cos(a+b) = \cos a \cos b - \sin a \sin b\): \(\cos (2 \pi / 3) = \cos(\pi/3) \cos(\pi/3) - \sin(\pi/3) \sin(\pi/3)\) Recall that: \(\cos(\pi/3) = 1/2\) and \(\sin(\pi/3) = \sqrt{3}/2\) So, substitute these values in the formula: \(\cos (2 \pi / 3) = (1/2)(1/2) - (\sqrt{3}/2)(\sqrt{3}/2)\) \(\cos (2 \pi / 3) = 1/4 - 3/4\) \(\cos (2 \pi / 3) = -1/2\)

Step 5: Final calculation of expression

Replace the value of \(\cos (2 \pi / 3)\) in the simplified expression: \((-1)\cos (2 \pi / 3)+0 = -1(-1/2) = 1/2\) The final value of the expression is \(\boxed{\frac{1}{2}}\).

Key Concepts

Cosine Addition Formula

The cosine addition formula is an important concept in trigonometry that allows us to simplify expressions involving the cosine of a sum of two angles. The formula is expressed as \(\cos(a + b) = \cos a \cos b - \sin a \sin b\). In our textbook exercise, you'll see how this formula transforms a complex trigonometric expression into something much simpler. To apply the formula, we first need to recognize that trigonometric functions have certain specific values at known angles, which in our case are angles that are multiples of \(\frac{\pi}{3}\).

For instance, if we look at the expression \(\cos(2 \pi / 3)\), we can interpret it as \(\cos(\pi/3 + \pi/3)\) and apply the cosine addition formula. Here’s how we do it:

• Firstly, we need to know the values for \(\cos(\pi/3)\) and \(\sin(\pi/3)\), which are respectively \(1/2\) and \(\sqrt{3}/2\).
• By substituting these values into the formula, we get \(\cos(2 \pi / 3) = \cos(\pi/3) \cos(\pi/3) - \sin(\pi/3) \sin(\pi/3)\).
• Doing the math, this simplifies to \(\cos(2 \pi / 3) = 1/4 - 3/4\), which equals \(\-1/2\).

By utilizing the cosine addition formula, we can easily find the value of trigonometric expressions that otherwise may seem daunting.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is a crucial skill for solving a wide variety of math problems. The key to simplification is to recognize and apply trigonometric identities and properties efficiently. Trigonometric identities, like the ones utilized in our example, provide a set of tools that help us to convert complex expressions into simpler or more usable forms.

In the given exercise, the process begins by recognizing the values of trigonometric functions at specific angles—\(\cos(\pi)\) and \(\sin(\pi)\), which are -1 and 0, respectively. With these identities, the original complex expression simplifies significantly since multiplying by zero will eliminate terms. From there, we can focus on the remaining non-zero terms and use identities like the cosine addition formula to further simplify.

• By replacing known values and using identities to combine and reduce the terms, the expression \(\cos(2 \pi / 3) \cos(\pi) + \sin(2 \pi / 3) \sin(\pi)\) simplifies to \(\-\cos(2 \pi / 3)\).
• Following our prior step with the cosine addition formula, we find that \(\-\cos(2 \pi / 3)\) simplifies to \(1/2\).

This clear, step-by-step approach allows students to methodically work through trigonometric expressions and reach a solution without getting overwhelmed.

Radian Measure

Radian measure is the standard unit of angular measurement used in mathematics. Unlike degrees, which break a circle into 360 arbitrary units, radians provide a measure that relates the arc length of a circle directly to its radius. One full revolution of a circle (all the way around) is \(2\pi\) radians, because the circumference of a circle is \(2\pi r\), where \(r\) is the radius.

In trigonometry, especially when dealing with trigonometric identities and simplifying expressions, it’s crucial to work in radians because many of the identities are rooted in radian measure. For instance, in our exercise, we're dealing with angles like \(2 \pi / 3\) and \(\pi\), which are in radian format. Without a proper understanding of how radians work, it can be challenging to visualize or memorize the values of trigonometric functions at these angles.

• Remembering that \(\pi\) radians is equivalent to 180 degrees, we can understand why \(\cos(\pi)\) is -1 since \(\cos(180^\circ)\) reflects the same value due to their equivalence.
• Similarly, \(2 \pi / 3\) radians corresponds to 120 degrees, helping us recognize familiar angles and the corresponding trigonometric values.

With frequent practice, converting between radians and degrees and applying this knowledge to simplify trigonometric expressions becomes a lot easier.