what is the exact value of sin (195)?

The exact value of sin(195) can be found using the unit circle or the sine function's periodic properties. Here are the steps:

1. Normalize the angle: The sine function has a period of 360 degrees, so we can add or subtract 360 degrees to any angle until we get an angle that is easier to work with. In this case, 195 - 180 = 15, so sin(195) = sin(15).

2. Use the sine addition formula: The sine addition formula is sin(a + b) = sin(a)cos(b) + cos(a)sin(b). We can use this formula with a = 45 degrees and b = 30 degrees, because 45 + 30 = 75, which is close to our desired angle of 15 degrees.

3. Calculate: sin(45)cos(30) + cos(45)sin(30) = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4.

So, sin(195) = sin(15) = (√6 + √2)/4.

The exact value of sin(195) can be found by using the unit circle or the sine function's periodic properties. Here are the steps:

1. Normalize the angle: The sine function has a period of 360 degrees, so we can add or subtract 360 degrees to any angle until we get an angle that is between 0 and 360 degrees. In this case, 195 degrees is already within this range, so no normalization is needed.

2. Find the reference angle: The reference angle is the acute angle that the given angle makes with the x-axis. For angles in the second quadrant (90 to 180 degrees) or third quadrant (180 to 270 degrees), the reference angle can be found by subtracting the given angle from 180 degrees. In this case, the reference angle is 180 - 195 = -15 degrees.

3. Find the sine of the reference angle: The sine of an angle in the unit circle is the y-coordinate of the point where the terminal side of the angle intersects the unit circle. For -15 degrees, this value is -sin(15 degrees).

4. Adjust for the quadrant: The sine function is positive in the first and second quadrants, and negative in the third and fourth quadrants. Since 195 degrees is in the third quadrant, the sine of 195 degrees is -sin(15 degrees).

5. Calculate or look up sin(15 degrees): The exact value of sin(15 degrees) is (√

The sine of an angle in the third quadrant (180° to 270°) is negative.

The reference angle for 195° is 195° - 180° = 15°.

The sine of 15° is √6 - √2 / 4.

Therefore, the sine of 195° is - (√6 - √2 / 4).