Introduction:
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is highly useful in various fields, such as physics, engineering, and navigation. One important trigonometric function is sine (sin), which calculates the ratio of the length of the side opposite an angle to the length of the hypotenuse. In this article, we’ll explore the exact value of sin 165 and answer some related FAQs.
What is the exact value of sin 165?
The exact value of sin 165 is **√(2 – √3) / 2**, which is approximately 0.2588.
Frequently Asked Questions:
1. What is sine?
Sine (sin) is a trigonometric function that relates the angles of a right triangle to the ratios of its sides.
2. How is sine calculated?
Sine can be calculated by dividing the length of the side opposite the given angle by the length of the hypotenuse in a right triangle.
3. What are the values of sin for common angles?
For commonly used angles, the values of sin are: sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, and sin 90° = 1.
4. How can you find the exact value of sin 165?
To find the exact value of sin 165, we can use various trigonometric identities and formulas.
5. Can sin 165 be expressed in simpler radicals?
No, the exact value of sin 165 cannot be expressed in simpler radicals without using trigonometric identities.
6. How is sin 165 derived?
The sine of 165 can be derived by using the sum-angle formula for the sine function or by utilizing the trigonometric identity for sin (180° – θ).
7. How can sin 165 be expressed in degrees?
Sin 165 is expressed in degrees as sin 165°.
8. What is the value of sin 180 – 15?
Since sin (180° – θ) = sin θ, the value of sin 180 – 15 is the same as sin 15, which is approximately 0.2588.
9. Is there a trigonometric table for sin 165?
Yes, trigonometric tables exist that list the approximate values of sine for various angles, including sin 165.
10. How can the value of sin 165 be verified?
The value of sin 165 can be verified using a scientific calculator, which often has a trigonometric function mode to calculate the value accurately.
11. What is the graph of sin 165?
The graph of sine function for sin 165 would resemble the graph of any other sine function but shifted 165 units to the right on the x-axis.
12. In which quadrants is sine positive?
Sine is positive in the first and second quadrants of the unit circle, which correspond to angles between 0° and 180°.