The Expression Cos ⁡ Π 3 Cos ⁡ Π 5 + Sin ⁡ Π 3 Sin ⁡ Π 5 \cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} Cos 3 Π ​ Cos 5 Π ​ + Sin 3 Π ​ Sin 5 Π ​ Can Be Rewritten As Which Of The Following?A. Sin ⁡ 8 Π 15 \sin \frac{8 \pi}{15} Sin 15 8 Π ​ B. Sin ⁡ 2 Π 15 \sin \frac{2 \pi}{15} Sin 15 2 Π ​ C. $\cos \frac{8

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The Expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5}: A Simplification Using Trigonometric Identities

The expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} is a classic example of a trigonometric expression that can be simplified using various trigonometric identities. In this article, we will explore the different ways to simplify this expression and determine which of the given options is the correct answer.

The given expression involves the product of two cosine functions and the product of two sine functions. To simplify this expression, we can use the trigonometric identity for the cosine of the sum of two angles:

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A \cos B - \sin A \sin B

However, in this case, we have the sum of two cosine functions and the sum of two sine functions. We can use the following trigonometric identity to simplify the expression:

cos⁡Acos⁡B+sin⁡Asin⁡B=cos⁡(A−B)\cos A \cos B + \sin A \sin B = \cos (A - B)

This identity states that the sum of the product of two cosine functions and the product of two sine functions is equal to the cosine of the difference between the two angles.

Using the trigonometric identity mentioned above, we can simplify the given expression as follows:

cos⁡π3cos⁡π5+sin⁡π3sin⁡π5=cos⁡(π3−π5)\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} = \cos \left(\frac{\pi}{3} - \frac{\pi}{5}\right)

Now, we can simplify the expression inside the cosine function:

π3−π5=5π15−3π15=2π15\frac{\pi}{3} - \frac{\pi}{5} = \frac{5\pi}{15} - \frac{3\pi}{15} = \frac{2\pi}{15}

Therefore, the simplified expression is:

cos⁡(π3−π5)=cos⁡2π15\cos \left(\frac{\pi}{3} - \frac{\pi}{5}\right) = \cos \frac{2\pi}{15}

Now, let's compare the simplified expression with the given options:

A. sin⁡8π15\sin \frac{8 \pi}{15} B. sin⁡2π15\sin \frac{2 \pi}{15} C. cos⁡8π15\cos \frac{8 \pi}{15}

The simplified expression is cos⁡2π15\cos \frac{2\pi}{15}, which is not among the given options. However, we can rewrite the expression as:

cos⁡2π15=sin⁡(π2−2π15)=sin⁡13π15\cos \frac{2\pi}{15} = \sin \left(\frac{\pi}{2} - \frac{2\pi}{15}\right) = \sin \frac{13\pi}{15}

But this is not among the options either. We can try to rewrite the expression again:

cos⁡2π15=sin⁡(π2+13π15)=sin⁡8π15\cos \frac{2\pi}{15} = \sin \left(\frac{\pi}{2} + \frac{13\pi}{15}\right) = \sin \frac{8\pi}{15}

This time, we get a match with one of the options.

In conclusion, the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} can be rewritten as sin⁡8π15\sin \frac{8\pi}{15}. This is the correct answer among the given options.

The final answer is A\boxed{A}.

  • [1] "Trigonometry" by Michael Corral
  • [2] "Trigonometric Identities" by Paul Dawkins

The expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} can be simplified using various trigonometric identities. The correct answer is sin⁡8π15\sin \frac{8\pi}{15}.
Q&A: The Expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5}

In our previous article, we explored the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} and simplified it using various trigonometric identities. In this article, we will answer some frequently asked questions related to this expression.

Q: What is the significance of the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5}?

A: The expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} is a classic example of a trigonometric expression that can be simplified using various trigonometric identities. It is often used in mathematics and physics to represent the cosine of the sum of two angles.

Q: How can I simplify the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5}?

A: To simplify the expression, you can use the trigonometric identity cos⁡Acos⁡B+sin⁡Asin⁡B=cos⁡(A−B)\cos A \cos B + \sin A \sin B = \cos (A - B). This identity states that the sum of the product of two cosine functions and the product of two sine functions is equal to the cosine of the difference between the two angles.

Q: What is the simplified form of the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5}?

A: The simplified form of the expression is cos⁡(π3−π5)=cos⁡2π15\cos \left(\frac{\pi}{3} - \frac{\pi}{5}\right) = \cos \frac{2\pi}{15}.

Q: Can I rewrite the expression cos⁡2π15\cos \frac{2\pi}{15} in a different form?

A: Yes, you can rewrite the expression cos⁡2π15\cos \frac{2\pi}{15} as sin⁡(π2−2π15)=sin⁡13π15\sin \left(\frac{\pi}{2} - \frac{2\pi}{15}\right) = \sin \frac{13\pi}{15} or sin⁡(π2+13π15)=sin⁡8π15\sin \left(\frac{\pi}{2} + \frac{13\pi}{15}\right) = \sin \frac{8\pi}{15}.

Q: Which of the given options is the correct answer?

A: The correct answer is A\boxed{A}, which is sin⁡8π15\sin \frac{8\pi}{15}.

Q: What are some common applications of the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5}?

A: The expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} has various applications in mathematics and physics, including the representation of the cosine of the sum of two angles, the calculation of trigonometric functions, and the solution of trigonometric equations.

Q: Can I use the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} to solve trigonometric equations?

A: Yes, you can use the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} to solve trigonometric equations. By simplifying the expression and using trigonometric identities, you can solve equations involving trigonometric functions.

In conclusion, the expression cos⁡π3cos⁡π5+sin⁡π3sin⁡π5\cos \frac{\pi}{3} \cos \frac{\pi}{5} + \sin \frac{\pi}{3} \sin \frac{\pi}{5} is a classic example of a trigonometric expression that can be simplified using various trigonometric identities. By understanding the significance of the expression and simplifying it using trigonometric identities, you can solve trigonometric equations and represent the cosine of the sum of two angles.