Introduction
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It has many applications in various fields such as physics, engineering, and astronomy. One common problem in trigonometry is finding the exact value of trigonometric functions for specific angles. In this article, we will focus on finding the exact value of sin(165°) and explore related frequently asked questions.
Finding the Exact Value of sin(165°)
To find the exact value of sin(165°), we can make use of a special angle relationship in trigonometry. We know that sin(180° – θ) = sin(θ). Therefore, sin(165°) is equal to sin(180° – 15°).
Answer: To find the exact value of sin(165°), we evaluate sin(180° – 15°).
To proceed, we use the identity sin(A – B) = sin(A)cos(B) – cos(A)sin(B). Set A = 180° and B = 15°:
sin(180° – 15°) = sin(180°)cos(15°) – cos(180°)sin(15°)
Frequently Asked Questions:
1. What is the exact value of cos(15°)?
The exact value of cos(15°) is (√6 + √2)/4.
2. What is the exact value of sin(180°)?
The exact value of sin(180°) is 0.
3. What is the exact value of cos(180°)?
The exact value of cos(180°) is -1.
4. What is the exact value of sin(15°)?
The exact value of sin(15°) is (√6 – √2)/4.
5. What is the exact value of sin(165°) in terms of cos(15°) and sin(15°)?
sin(165°) = 0 * cos(15°) – (-1) * sin(15°) = sin(15°).
6. What is the value of sin(15°)?
The value of sin(15°) is approximately 0.2588.
7. Is the exact value of sin(165°) rational or irrational?
The exact value of sin(165°), which is sin(15°), is irrational.
8. How can we find the exact value of cos(15°)?
One method to find the exact value of cos(15°) is by using a half-angle formula, but it involves using trigonometric identities and might be more complex.
9. Can the value of sin(165°) be expressed as a fraction or a radical?
Yes, the value of sin(165°) can be expressed exactly as (√6 – √2)/4, which is a difference of radicals.
10. Is sin(165°) a positive or negative value?
sin(165°) is a positive value because sin(15°) is positive.
11. What is the relationship between sin(θ) and cos(θ)?
The relationship between sin(θ) and cos(θ) is defined by the Pythagorean identity: sin(θ) = √(1 – cos²(θ)) and cos(θ) = √(1 – sin²(θ)).
12. Are there any practical applications for sin(165°)?
sin(165°) and trigonometric functions, in general, are widely used in various fields, including navigation, physics, and computer graphics, to solve real-world problems involving angles and distances.
Conclusion
Finding the exact value of trigonometric functions for specific angles is an important aspect of trigonometry. By using special angle relationships and trigonometric identities, we can determine the exact value of sin(165°) to be sin(15°). Trigonometry plays a crucial role in many practical applications, making it a valuable mathematical tool in numerous disciplines.