cos(-11pi/12)
sometimes I find it easier to visualize these kind of angles in degrees
π/12 radians = 15°,
so -11π/12 = -165°
and 165 = 120+45
cos(-11π/12) = cos(-165° = cos(165°)
= cos(120 + 45)
= cos120cos45 - sin120sin45
= (-1/2)(1/√2) - (√3/2)(1/√2)
= (-1 - √3)/(2√2)
or
= (-√2 - √6)/4
To calculate the value of cos(-11π/12), follow these steps:
Step 1: Begin by determining the reference angle. The reference angle is the positive angle between the terminal side of the given angle and the x-axis.
Step 2: Convert the given angle to the equivalent positive angle within the range of 0 to 2π (or 0 to 360 degrees) by adding or subtracting multiples of 2π (or 360 degrees) as needed. In this case, -11π/12 is already within this range.
Step 3: Determine the quadrant in which the angle falls:
- In the first quadrant, cosine is positive.
- In the second quadrant, cosine is negative.
- In the third quadrant, cosine is negative.
- In the fourth quadrant, cosine is positive.
Step 4: Using the reference angle and the quadrant determined in Step 3, evaluate the cosine function:
- If the angle falls in the first quadrant, the cosine value will be the same as the cosine of the reference angle.
- If the angle falls in the second quadrant, the cosine value will be negative of the cosine of the reference angle.
- If the angle falls in the third quadrant, the cosine value will be negative of the cosine of the reference angle.
- If the angle falls in the fourth quadrant, the cosine value will be the same as the cosine of the reference angle.
Step 5: Calculate the reference angle:
- Since the given angle is -11π/12, the reference angle will be π/12 (since 11π/12 + π/12 = π).
Step 6: Determine the quadrant:
- Since the angle lies in the third quadrant (between 180 degrees and 270 degrees), the cosine will be negative.
Step 7: Evaluate the cosine function using the reference angle:
- The cosine of π/12 is √6 - √2 / 4.
Step 8: Since the angle falls in the third quadrant (as determined in Step 6), the cosine value will be negative. Therefore, cos(-11π/12) = -√6 - √2 / 4.
Therefore, cos(-11π/12) = -√6 - √2 / 4.
To find the value of cosine of (-11π/12), we can use the unit circle or the periodicity property of cosine.
The unit circle is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. The angle measure is counterclockwise from the positive x-axis.
To find the value of cosine at any given angle on the unit circle, we can use the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
In this case, -11π/12 is a negative angle measured counterclockwise from the positive x-axis. Since cos(-(θ)) is equal to cos(θ), we can find the value of cos(-11π/12) by finding the cosine value of positive 11π/12.
To determine the x-coordinate of the unit circle at an angle of 11π/12, we need to consider the reference angle. The reference angle is the acute angle between the terminal side and the x-axis.
In this case, the reference angle is 11π/12 - π/2 = 11π/12 - 6π/12 = 5π/12.
The reference angle is within the first quadrant of the unit circle, where both the x and y coordinates are positive. Therefore, we can determine that cos(5π/12) is positive.
Thus, to find the value of cos(-11π/12):
1. Find the reference angle: 5π/12.
2. Determine the cosine value of the reference angle: cos(5π/12).
3. Since cos(θ) = cos(-θ), the value of cos(-11π/12) is equal to cos(11π/12), which is the cosine value of the reference angle obtained in step 2.
You can use a scientific calculator or reference tables to find the decimal approximation of cos(5π/12) or cos(11π/12).
Please note that the value of cos(-11π/12) will be the same as the value of cos(11π/12), but with opposite sign due to the negative angle.