The value of sine is a fundamental concept in trigonometry that helps us understand the relationship between angles and the lengths of the sides of a right-angled triangle. Sine is defined as the ratio of the length of the side opposite to an angle to the length of the hypotenuse of the triangle.
To determine the values of sine for various angles, we can use the sine formula: sin x = Perpendicular/Hypotenuse. This formula tells us that the sine of an angle is equal to the length of the side opposite to the angle divided by the length of the hypotenuse.
Let’s consider some specific angles and their corresponding sine values.
At angle 0 (0 radians or 0 degrees), the length of the side opposite to the angle is 0, and the length of the hypotenuse is non-zero. Therefore, sin 0 = 0/Non-zero = 0.
Moving on to angle π/6 (30 degrees), we can create a right-angled triangle where the length of the side opposite to the angle is equal to the length of the hypotenuse divided by 2. Therefore, sin π/6 = (1/2)/1 = 1/2.
At angle π/4 (45 degrees), we have a right-angled triangle where the length of the side opposite to the angle is equal to the length of the hypotenuse divided by the square root of 2. Hence, sin π/4 = (1/√2)/1 = 1/√2.
For angle π/3 (60 degrees), we construct a right-angled triangle where the length of the side opposite to the angle is equal to the length of the hypotenuse multiplied by the square root of 3 divided by 2. Therefore, sin π/3 = (√3/2)/1 = √3/2.
Lastly, at angle π/2 (90 degrees), the side opposite to the angle is the longest side, i.e., equal to the length of the hypotenuse. Thus, sin π/2 = 1/1 = 1.
These values of sine for specific angles are commonly used in trigonometry and have been determined through mathematical calculations and geometric principles. They are essential in solving various trigonometric problems and understanding the relationships between angles and sides in right-angled triangles.
It is worth noting that the sine values can also be negative for angles in the second and third quadrants of the coordinate system. However, in this explanation, we have focused on the positive values of sine for simplicity.
Understanding the values of sine allows us to calculate unknown sides or angles in right-angled triangles, as well as analyze and solve problems involving periodic functions and waveforms. Trigonometry, including the sine function, is used extensively in fields such as physics, engineering, and mathematics.
The value of sine varies depending on the angle in a right-angled triangle. The commonly used values of sine, such as sin 0, sin π/6, sin π/4, sin π/3, and sin π/2, have been determined using the sine formula, which relates the length of the side opposite to an angle to the length of the hypotenuse. These values are fundamental in trigonometry and have practical applications in various fields.