Find the exact values of the cosine and sine of the angle. Then find the decimal values. Show your work step by step.
0 = 330°
To find the exact values of the cosine and sine of the angle 330°, we need to work with reference angles within the range of 0° to 90°. Let's find the reference angle first:
The reference angle is found by subtracting the closest multiple of 360° from the given angle. In this case, the closest multiple of 360° to 330° is 360° itself, as it is the first multiple that is greater than 330°.
Reference angle = 330° - 360° = -30°
However, since we are working with angles between 0° and 90°, we need to consider an equivalent positive angle for -30° within this range.
Since the cosine function is positive in the first and fourth quadrants, and the sine function is positive in the first and second quadrants, we need to find the positive equivalent angles in these quadrants.
-30° is equivalent to 360° - 30° = 330° in the first quadrant and 360° + (-30°) = 330° in the fourth quadrant.
Now, let's find the exact values for cosine and sine separately:
1) For the angle 30° in the first quadrant:
cos(30°) = √3/2
sin(30°) = 1/2
2) For the angle 30° in the fourth quadrant:
cos(30°) = √3/2
sin(30°) = -1/2 (negative value because sine is negative in the fourth quadrant)
Now, let's find the decimal values:
1) For the angle 30° in the first quadrant:
cos(30°) ≈ 0.866
sin(30°) ≈ 0.5
2) For the angle 30° in the fourth quadrant:
cos(30°) ≈ 0.866
sin(30°) ≈ -0.5