Arccosine: Value
How to find the given arccosine value: formula, 1 example and its solution.
Formula
Formula
Arccosine is the inverse function of cosine.
So, to solve arccosine,
set x = arccos y,
write cos x = y,
and solve the cosine equation.
x is in (0 ≤ x ≤ π).
y = cos x is not one-to-one.
But if (0 ≤ x ≤ π),
y = cos x is one-to-one.
So its inverse function can be defined.
This is why x is in (0 ≤ x ≤ π).
Example
Example
Solution
set x = arccos (-1/2).
Then cos x = -1/2.
x is in (0 ≤ x ≤ π).
Draw a right triangle
that satisfies
cos x = -1/2 and (0 ≤ x ≤ π).
0 ≤ x ≤ π
So the right triangle should be in
either quadrant I or quadrant II.
See cos x = (-1/2).
Cosine is CAH:
Cosine,
Adjacent side (-1),
Hypotenuse (2).
So draw a right triangle on a coordinate plane
whose adjacent side is -1
and whose hypotenuse is 2.
Draw the angle x
that starts from the 3 o'clock position.
Find the missing side
by using the Pythagorean theorem:
(-1)2 + [height]2 = 22.
Then the height is √3.
This is a right triangle
whose sides are -1, √3, and 2.
So this is a 30-60-90 triangle.
So the central angle is, 60º, π/3.
Radian Measure
π/3 and x are supplementary.
So x = π - π/3.
So 2π/3 is the answer.