exact value of
cot[arccos(-2/3)]
To find the exact value of cot[arccos(-2/3)], we need to follow a step-by-step process.
Step 1: Find the value of arccos(-2/3)
To find the value of arccos(-2/3), we need to find the angle whose cosine is -2/3.
cos(theta) = -2/3
Step 2: Use the Pythagorean identity
To determine the value of cosine, we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
Since cosine is negative, we need to use the negative square root for sin(theta):
sin^2(theta) + (-2/3)^2 = 1
sin^2(theta) + 4/9 = 1
sin^2(theta) = 1 - 4/9
sin^2(theta) = 5/9
Step 3: Determine the value of sin(theta)
Taking the square root of both sides, we get:
sin(theta) = √(5/9)
Step 4: Determine the value of cot(theta)
The cotangent is the reciprocal of the tangent, and the tangent is the ratio of sine to cosine:
cot(theta) = 1/tan(theta)
Using the identity tan(theta) = sin(theta)/cos(theta), we can rewrite the expression as:
cot(theta) = 1 / (sin(theta)/cos(theta))
= cos(theta) / sin(theta)
Step 5: Plug in the values
Now, we substitute the value we found for sin(theta) and find the value of cos(theta):
sin(theta) = √(5/9)
To find cos(theta), we can use the Pythagorean identity again:
sin^2(theta) + cos^2(theta) = 1
(√(5/9))^2 + cos^2(theta) = 1
5/9 + cos^2(theta) = 1
cos^2(theta) = 1 - 5/9
cos^2(theta) = 4/9
Taking the square root of both sides, we get:
cos(theta) = ±√(4/9)
Since we know that cosine is negative in this case, we take the negative square root:
cos(theta) = -√(4/9) = -2/3
Substituting these values into the expression for cot(theta):
cot(theta) = cos(theta) / sin(theta)
= (-2/3) / (√(5/9))
= -2/3 * (√(9/5) / 3)
= (-2/3) * (√9 / √5)
= -2/3 * 3/√5
= -6 / √5
= (-6/√5) * (√5/√5) [Multiplying by √5/√5]
= -6√5 / 5
Therefore, the exact value of cot[arccos(-2/3)] is -6√5 / 5.