What is the exact value of sin 225

- sqrt of 2 over 2

To find the exact value of sin 225 degrees, we can use the unit circle and the periodicity of the sine function.

First, we need to convert 225 degrees into a form that is more easily related to the unit circle. Since 225 degrees is greater than 180 degrees, we can subtract 180 degrees to get an equivalent angle between 0 and 180 degrees.

225 degrees - 180 degrees = 45 degrees

Now we have an angle of 45 degrees, which is a well-known angle on the unit circle. The coordinates of the point corresponding to 45 degrees on the unit circle are (√2/2, √2/2) or approximately (0.707, 0.707).

Since sine is the y-coordinate of a point on the unit circle, the exact value of sin 225 degrees is √2/2 or approximately 0.707.

To find the exact value of sin 225 degrees, we need to use some trigonometric identities and the fact that the sine function is periodic.

First, let's recognize that 225 degrees is in the third quadrant of the unit circle. In the third quadrant, the sine function is negative. We can determine this because the x-coordinate (cosine) is positive, and the y-coordinate (sine) is negative.

Next, we need to find the reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. It is obtained by subtracting 180 degrees from the given angle in the third quadrant. So, the reference angle of 225 degrees is 225 - 180 = 45 degrees.

Now, let's recall the trigonometric identity for the sine of an angle larger than 90 degrees:
sin(180 - θ) = sin θ.

Using this identity, we can rewrite sin 225 degrees as sin (180 - 45) degrees, which is equal to sin 135 degrees.

The exact value of sin 135 degrees can be determined from the special angles in the unit circle. At 135 degrees, the coordinates on the unit circle are (-√2/2, √2/2). Since we are dealing with sine, which represents the y-coordinate, the exact value of sin 135 degrees is √2/2.

Therefore, the exact value of sin 225 degrees is -√2/2.

you know that sin 45 = √2/2

225 = 180+45, so
sin(225) = sin(180+45)
now just use your addition formula to get the above result.