I get that we have to put it into radical form, but I don't understand the process. Can someone walk me through it and explain it?
arccos(-1/2)
you want the angle Ø so that
cosØ= -1/2
We know that if the cosØ is negative, then Ø is in either the II or III quadrant
we also know that cos 60° =+1/2
so Ø = 180-60 or 180+60
Ø = 120° or 240°
in radians it would be 2π/3 or 4π/3
your calculator will only give you the angle closest to the origin,
inter
2nd
cos
-.5
=
and you will get 120 if your calculator is set to degrees.
(12x2.17)
cos0 120.6
To find the radical form of the given expression, arccos(-1/2), we need to understand the concept of inverse trigonometric functions and how they relate to the ratios of right triangles.
The arccos function, also known as the inverse cosine function, is the inverse of the cosine function. It takes an input between -1 and 1 (inclusive) and returns an angle measured in radians. If we have a right triangle, the arccos function tells us the angle whose cosine is equal to a given value.
In this case, we are trying to find the angle whose cosine is -1/2. To understand this, let's consider the unit circle, which has a radius of 1.
When the cosine of an angle is -1/2, it means that the x-coordinate of the point on the unit circle corresponding to that angle is -1/2. In other words, we are looking for the angle where the adjacent side of a right triangle is -1/2 units long, while the hypotenuse is 1 unit long.
To visualize this, draw a right triangle with one angle equal to the angle we are trying to find, and let the adjacent side be -1/2 and the hypotenuse be 1. Using the Pythagorean theorem, we can solve for the opposite side length.
Let's denote the opposite side by 'y'. According to the Pythagorean theorem, the sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse. So we have:
(-1/2)^2 + y^2 = 1^2
1/4 + y^2 = 1
y^2 = 1 - 1/4
y^2 = 3/4
Taking the square root of both sides, we get:
y = ±√(3/4)
Since the angle we are looking for is in the fourth quadrant where the y-coordinate is negative, the negative square root is the one we want:
y = -√(3/4)
Now, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is √4:
y = -√(3/4) * (√4/√4)
y = -√(3/4) * (√4/2)
y = -√(3/2)
So, the value of y is -√(3/2). Therefore, the angle (in radians) whose cosine is -1/2 is represented in radical form as arccos(-1/2) = -√(3/2).
In conclusion, the radical form of arccos(-1/2) is -√(3/2).