find exat value for cos 135 and sin 135
To find the exact values of cos 135 and sin 135, we can use the unit circle.
In the unit circle, the point corresponding to 135 degrees is located in the third quadrant. The angle formed by the point on the unit circle and the positive x-axis is 45 degrees (180 degrees - 135 degrees).
Since the point is in the third quadrant, both the x-coordinate and y-coordinate would be negative.
For cos(135), the x-coordinate of the point is the value we are looking for. Since the angle formed by the point and the positive x-axis is 45 degrees, cos(45) = 1 / √2. However, since the point is in the third quadrant, the x-coordinate would be negative. Therefore, cos(135) = -1 / √2 = -√2 / 2.
For sin(135), the y-coordinate of the point is the value we are looking for. Since sin is positive in the third quadrant, we can use the absolute value of the y-coordinate. Since the angle formed by the point and the positive x-axis is 45 degrees, sin(45) = 1 / √2. Therefore, sin(135) = 1 / √2 = √2 / 2.
So, cos 135 = -√2 / 2 and sin 135 = √2 / 2.
To find the exact values for cosine (cos) and sine (sin) of 135 degrees, we can use the values for common angles.
Step 1: Convert 135 degrees to a reference angle in the first quadrant.
Since 135 degrees is in the second quadrant, we subtract it from 180 degrees to get the equivalent angle in the first quadrant.
Reference angle = 180 degrees - 135 degrees = 45 degrees
Step 2: Determine the sign (+/-) of cosine and sine in the second quadrant.
In the second quadrant, cosine is negative, and sine is positive.
Step 3: Apply the values for the reference angle (45 degrees).
The cosine of 45 degrees is √2/2.
The sine of 45 degrees is √2/2.
Step 4: Apply the sign to cosine and sine in the second quadrant.
Since cosine is negative in the second quadrant:
cos 135 = - √2/2
Since sine is positive in the second quadrant:
sin 135 = √2/2
So, the exact values for cos 135 and sin 135 are:
cos 135 = - √2/2
sin 135 = √2/2