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Which of the following is the exact value of [tex]\cos (\pi) \cos \left(\frac{\pi}{3}\right) - \sin (\pi) \sin \left(\frac{\pi}{3}\right)[/tex]?

A. [tex]-\frac{1}{2}[/tex]
B. [tex]\frac{1}{2}[/tex]
C. [tex]-\frac{\sqrt{2}}{2}[/tex]
D. [tex]\frac{\sqrt{2}}{2}[/tex]

Sagot :

GinnyAnswer
To find the exact value of the expression [tex]\(\cos (\pi) \cos \left(\frac{\pi}{3}\right) - \sin (\pi) \sin \left(\frac{\pi}{3}\right)\)[/tex], let's break down the steps and evaluate each component individually.

First, recall the exact values of the trigonometric functions at specific angles:

1. [tex]\(\cos(\pi)\)[/tex]:
[tex]\[ \cos(\pi) = -1 \][/tex]

2. [tex]\(\cos\left(\frac{\pi}{3}\right)\)[/tex]:
[tex]\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \][/tex]

3. [tex]\(\sin(\pi)\)[/tex]:
[tex]\[ \sin(\pi) = 0 \][/tex]

4. [tex]\(\sin\left(\frac{\pi}{3}\right)\)[/tex]:
[tex]\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \][/tex]

Next, substitute these values into the expression:
[tex]\[ \cos (\pi) \cos \left(\frac{\pi}{3}\right) - \sin (\pi) \sin \left(\frac{\pi}{3}\right) \][/tex]

We need to evaluate each part of the expression:

[tex]\[ \cos (\pi) \cos \left(\frac{\pi}{3}\right) = (-1) \cdot \left(\frac{1}{2}\right) = -\frac{1}{2} \][/tex]

[tex]\[ \sin (\pi) \sin \left(\frac{\pi}{3}\right) = 0 \cdot \left(\frac{\sqrt{3}}{2}\right) = 0 \][/tex]

Now combine these results:

[tex]\[ -\frac{1}{2} - 0 = -\frac{1}{2} \][/tex]

Hence, the exact value of the expression [tex]\(\cos (\pi) \cos \left(\frac{\pi}{3}\right) - \sin (\pi) \sin \left(\frac{\pi}{3}\right)\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex].

So, the correct answer is:
[tex]\[ -\frac{1}{2} \][/tex]
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