Question

List three angles (in radian measure) that have a cosine of -1/2.

A) π/3
B) 2π/3
C) 4π/3
D) 5π/6

Answer 1

The three angles with a cosine of -1/2 are A) π/3, B) 5π/3, and C) 7π/3.

Step-by-step explanation:

The cosine function represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. To find angles with a cosine of -1/2, we look for angles where the x-coordinate (adjacent side) is negative and the y-coordinate (opposite side) is positive or negative. In the unit circle, the x-coordinate is represented by the cosine of an angle. The cosine of π/3 (60 degrees) is 1/2, so the cosine of 5π/3 is also 1/2 due to the periodic nature of the cosine function. However, cosine is negative in the second and third quadrants, hence the angles 5π/3 and 7π/3 (which is equivalent to -π/3) have a cosine of -1/2.

Therefore, the angles π/3, 5π/3, and 7π/3 have a cosine of -1/2. These angles correspond to the points on the unit circle where the x-coordinate is -1/2, meeting the criteria for a cosine of -1/2. The unit circle represents the relationship between angles and trigonometric functions, helping us identify angles with specific cosine values by examining their x-coordinates on the circle.

Answer 2

The angles with a cosine of -1/2 in radian measure are approximately 2.618 radians (π/2 + π/3), 4.712 radians (2π/3), and 7.069 radians (4π/3) with decimal place accuracy. So the correct options are B), C) and D).

Step-by-step explanation:

To find angles with a given trigonometric function value, we use the inverse trigonometric functions. In this case, we want to find angles with a cosine of -1/2. We can use the inverse cosine function, denoted as arccos, to find these angles.

First, let's consider the range of values for the inverse cosine function. The function returns values in the range of [0, π] radians for values of cosine between -1 and 1. For cosine values less than -1 or greater than 1, we need to add or subtract multiples of 2π radians to bring the angle into this range.

To find angles with a cosine of -1/2, we can use a calculator or a mathematical software package to find the arccosine of -1/2. The results are approximately 2.618 radians (π/2 + π/3), 4.712 radians (2π/3), and 7.069 radians (4π/3) with decimal place accuracy. These angles correspond to points on the unit circle where the y-coordinate is -1/2 and the x-coordinate is the square root of 3/2 (since cosine is equal to y-coordinate divided by x-coordinate).

In summary, these angles represent points on the unit circle that have a cosine value of -1/2, which is equivalent to having an x-coordinate that is greater than their y-coordinate by a factor of approximately sqrt(3)/2. These angles can be useful in various mathematical applications, such as finding the position of celestial bodies or understanding wave patterns in physics and engineering. Therefore the correct options are B), C) and D).