Use the triangle to find sin(23), cos(23), sin(), cos() 25 24 B
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.2: Trigonometric Functions Of Angles
Problem 18E
Related questions
Question
![---
### Trigonometric Identities and Exact Values
#### 6. Use the triangle to find sin(2β), cos(2β), sin(β/2), cos(β/2)
In the given problem, we have a right triangle with the following dimensions:
- Hypotenuse (opposite the right angle) = 25 units
- Adjacent side to angle β = 24 units
- Opposite side to angle β = a units
Since it's a right triangle, we can use the Pythagorean theorem to find the value of \( a \):
\[ a^2 + 24^2 = 25^2 \]
\[ a^2 + 576 = 625 \]
\[ a^2 = 49 \]
\[ a = 7 \]
Using these values:
\[ \sin(\beta) = \frac{opposite}{hypotenuse} = \frac{7}{25} \]
\[ \cos(\beta) = \frac{adjacent}{hypotenuse} = \frac{24}{25} \]
Using double-angle formulas and half-angle formulas:
- **Double Angle Formulas**
\[ \sin(2\beta) = 2 \sin(\beta) \cos(\beta) \]
\[ \sin(2\beta) = 2 \left( \frac{7}{25} \right) \left( \frac{24}{25} \right) = \frac{2 \times 7 \times 24}{625} = \frac{336}{625} \]
\[ \cos(2\beta) = \cos^2(\beta) - \sin^2(\beta) \]
\[ \cos(2\beta) = \left( \frac{24}{25} \right)^2 - \left( \frac{7}{25} \right)^2 \]
\[ \cos(2\beta) = \frac{576}{625} - \frac{49}{625} = \frac{527}{625} \]
- **Half Angle Formulas**
\[ \sin\left(\frac{\beta}{2}\right) = \sqrt{\frac{1 - \cos(\beta)}{2}} \]
\[ \cos\left(\frac{\beta}{2}\right) = \sqrt{\frac{1 + \cos(\beta)}{2}} \]
#### 7. Write](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6841f1bf-ebcf-44b9-bde6-b1a56299f544%2F491ce828-1633-496a-a4f0-ec8867c9d5d3%2F8rtior9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Trigonometric Identities and Exact Values
#### 6. Use the triangle to find sin(2β), cos(2β), sin(β/2), cos(β/2)
In the given problem, we have a right triangle with the following dimensions:
- Hypotenuse (opposite the right angle) = 25 units
- Adjacent side to angle β = 24 units
- Opposite side to angle β = a units
Since it's a right triangle, we can use the Pythagorean theorem to find the value of \( a \):
\[ a^2 + 24^2 = 25^2 \]
\[ a^2 + 576 = 625 \]
\[ a^2 = 49 \]
\[ a = 7 \]
Using these values:
\[ \sin(\beta) = \frac{opposite}{hypotenuse} = \frac{7}{25} \]
\[ \cos(\beta) = \frac{adjacent}{hypotenuse} = \frac{24}{25} \]
Using double-angle formulas and half-angle formulas:
- **Double Angle Formulas**
\[ \sin(2\beta) = 2 \sin(\beta) \cos(\beta) \]
\[ \sin(2\beta) = 2 \left( \frac{7}{25} \right) \left( \frac{24}{25} \right) = \frac{2 \times 7 \times 24}{625} = \frac{336}{625} \]
\[ \cos(2\beta) = \cos^2(\beta) - \sin^2(\beta) \]
\[ \cos(2\beta) = \left( \frac{24}{25} \right)^2 - \left( \frac{7}{25} \right)^2 \]
\[ \cos(2\beta) = \frac{576}{625} - \frac{49}{625} = \frac{527}{625} \]
- **Half Angle Formulas**
\[ \sin\left(\frac{\beta}{2}\right) = \sqrt{\frac{1 - \cos(\beta)}{2}} \]
\[ \cos\left(\frac{\beta}{2}\right) = \sqrt{\frac{1 + \cos(\beta)}{2}} \]
#### 7. Write
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning