IDNLearn.com is committed to providing high-quality answers to your questions. Find in-depth and trustworthy answers to all your questions from our experienced community members.
Sagot :
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]
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]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.