Find the exact value of sin (195degrees)

To find the exact value of sin(195 degrees), we can use the unit circle and trigonometric identities.

Step 1: Convert the angle to radians.
Since the unit circle uses radians, we need to convert 195 degrees to radians.
To convert degrees to radians, we use the formula:
radians = degrees * π/180
In this case, radians = 195 * π/180 = (13π/12) radians.

Step 2: Find the reference angle.
The reference angle is the smallest angle between the terminal side of the angle and the positive x-axis on the unit circle.
To find the reference angle for 13π/12 radians, we can subtract the angle from the nearest full rotation, which is 2π radians.
Reference angle = 2π - (13π/12) = (24π/12) - (13π/12) = (11π/12) radians.

Step 3: Determine the sign of sin.
From the unit circle, we know that sin is positive in the first and second quadrants. Since 195 degrees is in the third quadrant, the sine value will be negative.

Step 4: Use the reference angle to find the exact value.
Using a special right triangle or memorized values, we can find that sin(11π/12 radians) = -(√(1 - cos(11π/12)²)).
To find cos(11π/12 radians), we can use the Pythagorean identity sin²θ + cos²θ = 1. Since sin(11π/12) is already known, we can solve for cos(11π/12) and substitute it into the formula.

Let's calculate it:

cos²(11π/12) = 1 - sin²(11π/12)
cos²(11π/12) = 1 - (sin(11π/12))²
cos²(11π/12) = 1 - (√(1 - cos(11π/12)²))²
cos²(11π/12) = 1 - 1 + cos(11π/12)²
0 = cos(11π/12)² - cos(11π/12)²
0 = 0

From this equation, we can see that cos(11π/12) = 0.

Now we can substitute the known values back into the original equation:

sin(195 degrees) = -(√(1 - cos(11π/12)²))
sin(195 degrees) = -√(1 - 0²)
sin(195 degrees) = -√(1 - 0)
sin(195 degrees) = -√1
sin(195 degrees) = -1

Therefore, the exact value of sin(195 degrees) is -1.