The Ultimate Guide: How to Calculate the Sine of 45 Degrees Using the Unit Circle and Special Triangles

sin 45°

The sine of 45 degrees can be calculated using the unit circle or special triangles

The sine of 45 degrees can be calculated using the unit circle or special triangles.

Using the unit circle:
The unit circle is a circle with a radius of 1 unit and its center at the origin (0, 0) on the coordinate plane. To find the sine of an angle, draw a line from the origin to the point on the unit circle that corresponds to the angle measured in degrees, and then find the y-coordinate of that point.

In the case of 45 degrees, the point on the unit circle will be located where the line from the origin forms a 45-degree angle with the positive x-axis. This point can be found by dividing the unit circle into quadrants. Since 45 degrees is in the first quadrant, the x and y coordinates of this point will be the same, equal to √2/2. Therefore, the sine of 45 degrees is √2/2.

Using special triangles:
Another way to find the sine of 45 degrees is by considering special triangles. In particular, a 45-45-90 triangle has two equal angles of 45 degrees, and the sides of this triangle follow a specific ratio.
In a 45-45-90 triangle, the sides are in the ratio 1:1:√2.

Since the sine of an angle is defined as the length of the side opposite the angle divided by the hypotenuse, for a 45-45-90 triangle, the side opposite the 45-degree angle is equal to the length of one of the legs, and the hypotenuse is equal to the length of the other leg.

Therefore, in a 45-45-90 triangle, the sine of 45 degrees is equal to the side opposite the angle (which is also a leg) divided by the hypotenuse. Using the ratio 1:1:√2, we can conclude that the sine of 45 degrees is 1/√2, which can be rationalized to √2/2.

Hence, both methods yield the same result: the sine of 45 degrees is √2/2.

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