exact value of
cot[arccos(-2/3)]
To find the exact value of cot[arccos(-2/3)], we need to simplify the expression step by step. Let's break it down:
1. Start with arccos(-2/3):
- The arccos function returns the angle whose cosine is a given value.
- arccos(-2/3) represents the angle θ whose cosine is -2/3.
2. Next, evaluate the cotangent (cot) of that angle:
- The cotangent function returns the ratio of the adjacent side to the opposite side of a right triangle.
- cot(θ) = adjacent side / opposite side.
Now, let's proceed further:
3. Draw a right triangle with an angle θ such that cos(θ) = -2/3:
- Since cosine is negative, we know that the angle θ lies in the second or third quadrant of the unit circle.
- In the unit circle, draw a right triangle such that its adjacent side is -2 and its hypotenuse is 3.
.
|\
| \
| \
| \
-2 \
| \
|______\
3
4. Now, we can calculate the opposite side using the Pythagorean theorem:
- Pythagorean theorem: a^2 + b^2 = c^2, where c represents the hypotenuse, and a and b represent the other two sides.
- In this case, the hypotenuse is 3, one side (adjacent) is -2, so we can solve for the opposite side as follows:
b^2 = c^2 - a^2
b^2 = 3^2 - (-2)^2
b^2 = 9 - 4
b^2 = 5
b = √5
5. Now that we know the lengths of the three sides of the triangle, we can find cot(θ):
- cot(θ) = adjacent side / opposite side = -2 / √5
- However, we usually rationalize the denominator by multiplying both the numerator and denominator by √5:
cot(θ) = (-2 / √5) * (√5 / √5)
= -2√5 / 5
Therefore, the exact value of cot[arccos(-2/3)] is -2√5 / 5.
cos(θ)=-2/3
So θ is in the second or third quadrant. There is not enough information to be sure of which.
adjacent = -2
hypotenuse = 3
opposite = ±√(3²-(-2)²)
= ±√(5)
Therefore
cot(θ)
=adjacent/opposite
=-2/(±√(5))
= ±(2/√(5))
=±(2√(5)/5)