What is Sin 45 degrees?
The Correct Answer and Explanation is :
The sine of 45 degrees, denoted as (\sin(45^\circ)), is equal to (\frac{\sqrt{2}}{2}) or approximately 0.7071. This value can be derived from the properties of a right triangle and the unit circle.
Explanation
- Understanding Sine: The sine function is a fundamental trigonometric function defined for angles in a right triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse. For a right triangle with angle (A), the sine is given by:
[
\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}
] - 45-Degree Angle: In a right triangle where one angle is 45 degrees, the other non-right angle must also be 45 degrees (since the angles in a triangle add up to 180 degrees). This creates an isosceles right triangle where the two legs are of equal length. If we denote the lengths of these legs as (x), the hypotenuse (h) can be found using the Pythagorean theorem:
[
h = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2}
] - Calculating Sine: Using the definition of sine:
[
\sin(45^\circ) = \frac{x}{h} = \frac{x}{x\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
] - Unit Circle: In the unit circle (a circle with a radius of 1), an angle of 45 degrees corresponds to the coordinates ((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})). The sine of the angle is represented by the y-coordinate of this point, confirming that (\sin(45^\circ) = \frac{\sqrt{2}}{2}).
- Applications: The sine function is widely used in various fields such as physics, engineering, and computer graphics. Knowing the sine values of common angles (like 0°, 30°, 45°, 60°, and 90°) is crucial for solving problems related to waves, oscillations, and rotations.
In summary, (\sin(45^\circ) = \frac{\sqrt{2}}{2}) is a critical value in trigonometry, rooted in the geometric properties of triangles and the unit circle.