Use an addition or subtraction formula to find the exact value in simplest form. Rationalize your denominator, if necessary. G 35π 5π 35π + sin 5T sin 0/6 18 18 18 COS COS 00 X
Use an addition or subtraction formula to find the exact value in simplest form. Rationalize your denominator, if necessary. G 35π 5π 35π + sin 5T sin 0/6 18 18 18 COS COS 00 X
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 59E
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![**Title: Solving Trigonometric Expressions Using Addition and Subtraction Formulas**
**Objective:**
Learn how to use addition or subtraction formulas to find the exact value of trigonometric expressions in the simplest form. Rationalize the denominator if necessary.
**Problem:**
Use an addition or subtraction formula to find the exact value in simplest form. Rationalize your denominator, if necessary.
**Expression:**
\[ \cos\left(\frac{35\pi}{18}\right) \cos\left(\frac{5\pi}{18}\right) + \sin\left(\frac{35\pi}{18}\right) \sin\left(\frac{5\pi}{18}\right) \]
**Solution:**
To solve this, we can use the angle addition formula for cosine, given as:
\[ \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \]
However, note that the given expression resembles the form:
\[ \cos(A)\cos(B) + \sin(A)\sin(B) = \cos(A - B) \]
So, to proceed, we set \( A = \frac{35\pi}{18} \) and \( B = \frac{5\pi}{18} \). According to the formula:
\[ \cos\left(\frac{35\pi}{18} - \frac{5\pi}{18}\right) = \cos\left(\frac{30\pi}{18}\right) = \cos\left(\frac{5\pi}{3}\right) \]
Now, simplify \( \cos\left(\frac{5\pi}{3}\right) \).
Since \( \frac{5\pi}{3} = 2\pi - \frac{\pi}{3} \),
\[ \cos\left(\frac{5\pi}{3}\right) = \cos\left(2\pi - \frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) \]
Finally, since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \),
\[ \cos\left(\frac{5\pi}{3}\right) = \frac{1}{2} \]
Therefore, the exact value in simplest form is:
\[ \boxed{\frac{1}{2}}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe92e694a-8dce-4f5c-a0b3-7db66cedb1e5%2Ff416a207-3fb4-4ee4-a9d6-f792e152ab0c%2Fed3r93q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Trigonometric Expressions Using Addition and Subtraction Formulas**
**Objective:**
Learn how to use addition or subtraction formulas to find the exact value of trigonometric expressions in the simplest form. Rationalize the denominator if necessary.
**Problem:**
Use an addition or subtraction formula to find the exact value in simplest form. Rationalize your denominator, if necessary.
**Expression:**
\[ \cos\left(\frac{35\pi}{18}\right) \cos\left(\frac{5\pi}{18}\right) + \sin\left(\frac{35\pi}{18}\right) \sin\left(\frac{5\pi}{18}\right) \]
**Solution:**
To solve this, we can use the angle addition formula for cosine, given as:
\[ \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \]
However, note that the given expression resembles the form:
\[ \cos(A)\cos(B) + \sin(A)\sin(B) = \cos(A - B) \]
So, to proceed, we set \( A = \frac{35\pi}{18} \) and \( B = \frac{5\pi}{18} \). According to the formula:
\[ \cos\left(\frac{35\pi}{18} - \frac{5\pi}{18}\right) = \cos\left(\frac{30\pi}{18}\right) = \cos\left(\frac{5\pi}{3}\right) \]
Now, simplify \( \cos\left(\frac{5\pi}{3}\right) \).
Since \( \frac{5\pi}{3} = 2\pi - \frac{\pi}{3} \),
\[ \cos\left(\frac{5\pi}{3}\right) = \cos\left(2\pi - \frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) \]
Finally, since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \),
\[ \cos\left(\frac{5\pi}{3}\right) = \frac{1}{2} \]
Therefore, the exact value in simplest form is:
\[ \boxed{\frac{1}{2}}
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