### Trigonometry Problem: Exact Value Calculation **Problem Statement:** Find the exact value of \( 2 \sin 75^\circ \cos 75^\circ \). **Multiple Choice Options:** 1. \( \boxed{-\frac{1}{2}} \) 2. \( \boxed{\frac{\sqrt{3}}{2}} \) 3. \( \boxed{-\frac{\sqrt{3}}{2}} \) 4. \( \boxed{\frac{1}{2}} \) To solve this problem, we can use the double-angle identity for sine: \[ 2 \sin A \cos A = \sin 2A \] Let's apply this identity to our given expression: \[ 2 \sin 75^\circ \cos 75^\circ = \sin (2 \cdot 75^\circ) = \sin 150^\circ \] Now, we need to find the value of \( \sin 150^\circ \). Since \( \sin 150^\circ = \sin (180^\circ - 30^\circ) = \sin 30^\circ \): \[ \sin 150^\circ = \sin 30^\circ \] From trigonometric tables or knowledge of special angles, we know: \[ \sin 30^\circ = \frac{1}{2} \] Thus: \[ \sin 150^\circ = \frac{1}{2} \] So, the exact value of \( 2 \sin 75^\circ \cos 75^\circ \) is: \[ \frac{1}{2} \] Therefore, the correct option is: 4. \( \boxed{\frac{1}{2}} \)
### Trigonometry Problem: Exact Value Calculation **Problem Statement:** Find the exact value of \( 2 \sin 75^\circ \cos 75^\circ \). **Multiple Choice Options:** 1. \( \boxed{-\frac{1}{2}} \) 2. \( \boxed{\frac{\sqrt{3}}{2}} \) 3. \( \boxed{-\frac{\sqrt{3}}{2}} \) 4. \( \boxed{\frac{1}{2}} \) To solve this problem, we can use the double-angle identity for sine: \[ 2 \sin A \cos A = \sin 2A \] Let's apply this identity to our given expression: \[ 2 \sin 75^\circ \cos 75^\circ = \sin (2 \cdot 75^\circ) = \sin 150^\circ \] Now, we need to find the value of \( \sin 150^\circ \). Since \( \sin 150^\circ = \sin (180^\circ - 30^\circ) = \sin 30^\circ \): \[ \sin 150^\circ = \sin 30^\circ \] From trigonometric tables or knowledge of special angles, we know: \[ \sin 30^\circ = \frac{1}{2} \] Thus: \[ \sin 150^\circ = \frac{1}{2} \] So, the exact value of \( 2 \sin 75^\circ \cos 75^\circ \) is: \[ \frac{1}{2} \] Therefore, the correct option is: 4. \( \boxed{\frac{1}{2}} \)
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter1: The Six Trigonometric Functions
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Problem 91PS: Draw an angle in standard position whose terminal side contains the point (3, –2). Find the...
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![### Trigonometry Problem: Exact Value Calculation
**Problem Statement:**
Find the exact value of \( 2 \sin 75^\circ \cos 75^\circ \).
**Multiple Choice Options:**
1. \( \boxed{-\frac{1}{2}} \)
2. \( \boxed{\frac{\sqrt{3}}{2}} \)
3. \( \boxed{-\frac{\sqrt{3}}{2}} \)
4. \( \boxed{\frac{1}{2}} \)
To solve this problem, we can use the double-angle identity for sine:
\[ 2 \sin A \cos A = \sin 2A \]
Let's apply this identity to our given expression:
\[ 2 \sin 75^\circ \cos 75^\circ = \sin (2 \cdot 75^\circ) = \sin 150^\circ \]
Now, we need to find the value of \( \sin 150^\circ \). Since \( \sin 150^\circ = \sin (180^\circ - 30^\circ) = \sin 30^\circ \):
\[ \sin 150^\circ = \sin 30^\circ \]
From trigonometric tables or knowledge of special angles, we know:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus:
\[ \sin 150^\circ = \frac{1}{2} \]
So, the exact value of \( 2 \sin 75^\circ \cos 75^\circ \) is:
\[ \frac{1}{2} \]
Therefore, the correct option is:
4. \( \boxed{\frac{1}{2}} \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd95c06f-a3ea-4120-9338-f910c4caee1b%2F9dedb33d-a76b-4528-823f-32032cbd29fd%2Ftc95srg.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometry Problem: Exact Value Calculation
**Problem Statement:**
Find the exact value of \( 2 \sin 75^\circ \cos 75^\circ \).
**Multiple Choice Options:**
1. \( \boxed{-\frac{1}{2}} \)
2. \( \boxed{\frac{\sqrt{3}}{2}} \)
3. \( \boxed{-\frac{\sqrt{3}}{2}} \)
4. \( \boxed{\frac{1}{2}} \)
To solve this problem, we can use the double-angle identity for sine:
\[ 2 \sin A \cos A = \sin 2A \]
Let's apply this identity to our given expression:
\[ 2 \sin 75^\circ \cos 75^\circ = \sin (2 \cdot 75^\circ) = \sin 150^\circ \]
Now, we need to find the value of \( \sin 150^\circ \). Since \( \sin 150^\circ = \sin (180^\circ - 30^\circ) = \sin 30^\circ \):
\[ \sin 150^\circ = \sin 30^\circ \]
From trigonometric tables or knowledge of special angles, we know:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus:
\[ \sin 150^\circ = \frac{1}{2} \]
So, the exact value of \( 2 \sin 75^\circ \cos 75^\circ \) is:
\[ \frac{1}{2} \]
Therefore, the correct option is:
4. \( \boxed{\frac{1}{2}} \)
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