find exact value of cos(23pi/6)
did you notice that 23π/6 is π/6 short of 2π ?
so and angle of 23π/6 is coterminal with -π/6
so cos(23π/6) = cos(-π/6) = cos(π/6)
from the 30-60-90 triangle ratios you should know that cos 30° = √3/2 ...... (30° = π/6 radians)
then cos 23π/6 = √3/2
To find the exact value of cosine (cos) of an angle, we can use the unit circle.
First, let's convert the angle to radians. We have:
23π/6 = 6π + π/6
Since 2π = 12π/6, we can rewrite the angle as:
23π/6 = 12π/6 + π/6 = 2π + π/6
On the unit circle, an angle of π/6 corresponds to a point that forms an equilateral triangle with the x-axis.
The cosine value for π/6 is √3/2.
Since π/6 equals 30 degrees, 2π is equivalent to a full revolution or 360 degrees. Therefore, the cosine value for 2π is also √3/2.
Thus, we can find the exact value of cos(23π/6) as:
cos(23π/6) = cos(2π + π/6) = cos(2π) * cos(π/6) - sin(2π) * sin(π/6)
cos(23π/6) = (√3/2) * (√3/2) - (0) * (1/2) = 3/4
Therefore, the exact value of cos(23π/6) is 3/4.
To find the exact value of cosine function at an angle of 23π/6, we can use the unit circle or reference angles.
1. Start by converting the angle to a reference angle within the range of 0 to 2π. In this case, we need to rotate the angle counterclockwise until the terminal side is between 0 to 2π.
23π/6 = (3 * 2π + π/6)
Therefore, the reference angle is π/6.
2. Identify the quadrant where the angle terminates. In this case, the terminal side of π/6 lies in the second quadrant.
3. Since cosine is negative in the second quadrant, we can determine the exact value of cosine by using the cosine of the reference angle.
The cosine of π/6 is √3/2.
4. However, since the angle is in the second quadrant, the cosine value will be negative.
Thus, the exact value of cos(23π/6) is -√3/2.
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