What is the value of sin 45 degrees?
**The value of sin 45 degrees is 0.70710678118.**
Sin 45 degrees, also written as sin(pi/4) or sin(π/4), is a commonly used trigonometric function in mathematics. It represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. In this case, the angle is 45 degrees, where each of the other two angles is 45 degrees as well, making it an isosceles right triangle.
To find the value of sin 45 degrees, it is useful to construct a right-angled triangle with one angle measuring 45 degrees. By careful examination, we can determine that the two acute angles of an isosceles right triangle are each 45 degrees. The hypotenuse (the side opposite the right angle) has a length of 1 unit, and each of the other two sides has a length of 1/√2 units.
Using the definition of the sine function, sin(theta) = opposite/hypotenuse, we can substitute the values of the opposite side (1/√2) and the hypotenuse (1) to find sin 45 degrees. Evaluating the expression yields:
sin 45 degrees = (1/√2) / 1 = 1/√2.
To simplify this result, we multiply both the numerator and denominator by √2:
sin 45 degrees = (1/√2) * (√2/√2) = √2 / 2.
Rationalizing the denominator (multiplying both the numerator and denominator by √2) gives us:
sin 45 degrees = (√2 / 2) * (√2 / √2) = 2 / 2√2 = 1 / √2.
To further simplify the result, we multiply both the numerator and denominator by √2 once again:
sin 45 degrees = (1 / √2) * (√2 / √2) = √2 / 2.
Thus, the value of sin 45 degrees is 0.70710678118, rounded to 11 decimal places.
FAQs:
1. What is a right-angled triangle?
A right-angled triangle is a triangle that contains one angle measuring exactly 90 degrees (a right angle).
2. What does sin(theta) represent?
The sine function, sin(theta), represents the ratio of the length of the side opposite the angle theta to the length of the hypotenuse in a right-angled triangle.
3. Can you provide an example of an isosceles right triangle?
Sure! An isosceles right triangle could have two shorter sides of length 1 and the hypotenuse of length √2.
4. How do you calculate the value of sin 45 degrees?
To calculate sin 45 degrees, you can set up a right-angled triangle with one angle measuring 45 degrees and find the ratio of the length of the opposite side to the hypotenuse.
5. Can sin 45 degrees be expressed as a fraction?
Yes, sin 45 degrees can be expressed as √2 / 2.
6. What is the exact value of sin 45 degrees?
The exact value of sin 45 degrees is √2 / 2.
7. Is sin 45 degrees a positive or a negative value?
sin 45 degrees is a positive value because the opposite side in the right-angled triangle is positive in this case.
8. Can you find sin 45 degrees using a calculator?
Yes, you can use a calculator to find the approximate value of sin 45 degrees, which is 0.70710678118.
9. What are the values of sin for other common angles?
Some other common sine values include sin 0 degrees = 0, sin 30 degrees = 0.5, sin 60 degrees = √3 / 2, and sin 90 degrees = 1.
10. What are the reciprocal values of sin 45 degrees?
The reciprocal value of sin 45 degrees, also known as the cosecant of 45 degrees, is √2.
11. What is the inverse of the sine of 0.70710678118?
The inverse sine (sin^-1) of 0.70710678118 is 45 degrees.
12. How is sin 45 degrees used in real-life applications?
Sin 45 degrees is often used in engineering, architecture, and physics to calculate angles and distances in various applications, such as trusses and bridge designs.