How do you solve -6[cot(2pi/3)-cot(pi/3)]?
You should have available to you :
- either in your notebook
-or in your text
- or memorized ,
the ratio of sides for the 30-60-90 as well as the 45-45-90 right-angled triangles
you should also have memorized the simple conversion from degrees to radians of the following
30° = π/6 radians
45° = π/4 radians
60° = π/3 radians
90° = π/2 radians
180° = π radians
so pulling apart your expression
cot(2π/3)
= 1/tan(120°)
but tan(120°) = -tan60° = -√3
cot(2π/3) = -1/√3
cot(π/3)
= 1/tan(60)
= 1/√3
-6[cot(2pi/3)-cot(pi/3)]
= -6[-1/√3 - 1/√3]
= -6(-2√3)
= 12/√3
= 12/√3 * √3/√3
= 12√3/3
= 4√3
btw, we did not "solve" the expression , we evaluated the expression. We solve equations, and there was no equation.
To solve the expression -6[cot(2π/3)-cot(π/3)], we can simplify each cotangent term and then perform the subtraction.
Step 1: Simplify the cotangent terms
The cotangent function can be simplified using the reciprocal identity: cot(x) = 1/tan(x).
So, cot(2π/3) = 1/tan(2π/3) and cot(π/3) = 1/tan(π/3).
Now, we need to find the tangent of 2π/3 and π/3.
Step 2: Find the tangent
The tangent function for an angle can be found using the sine and cosine values using the identity: tan(x) = sin(x)/cos(x).
For 2π/3, the sine value is √3/2 (from the unit circle) and the cosine value is -1/2 (from the unit circle). Therefore, tan(2π/3) = (√3/2) / (-1/2) = -√3.
For π/3, the sine value is √3/2 and the cosine value is 1/2. Therefore, tan(π/3) = (√3/2) / (1/2) = √3.
Now, substitute these values back into the expression:
-6[(1/(-√3)) - (1/√3)]
Step 3: Simplify the expression
To simplify the expression, we need to find a common denominator for both terms.
The common denominator is √3 × (-√3) = -3.
Rewriting the expression:
-6[(-√3)/(-3) - √3/(-3)]
Simplifying further:
-6[√3/3 + √3/3]
Combine like terms:
-6(2√3/3)
Finally, simplify:
-12√3/3 = -4√3
Therefore, the solution to -6[cot(2π/3)-cot(π/3)] is -4√3.