Question

Find cot 4pi/3

Please need it asap!!!!!

Answer 1

The point t on the unit circle is ( -1/2, √3 /2 ).

Explanation:

The intersection of the x-axis (the horizontal number line) and the y-axis in the cartesian system divides the coordinate plane into four equal pieces. Because each of these four areas occupies a quarter of the entire coordinate plane, they are collectively referred to as quadrants.

Consider the point t = - 4π/3 on the circle.

Now, the reference angle is π / 3.

- 4π / 3 is in quadrant 2 where the value of cosine is negative and sine is positive.

Therefore, the point t(x,y) will be:

( x, y ) = ( - cos ( π / 3 ), sin ( π / 3 ) )

( x, y ) = ( - 1/ 2 , (√3 )/ 2 )

Answer 2

`\cot\left((4\pi)/(3)\right)=(√(3))/(3)`

Explanation:

To find the exact value of cot(4π/3), we can use the cotangent identity and the unit circle.

The cotangent identity expresses the cotangent of an angle θ as the ratio of the cosine to the sine of that angle:

`\boxed{\begin{array}{c}\underline{\sf Cotangent\;identity}\\\\\cot \theta=(\cos \theta)/(\sin \theta)\end{array}}`

Therefore, cot(4π/3) can be expressed as:

`\cot\left((4\pi)/(3)\right) = (\cos\left((4\pi)/(3)\right))/(\sin\left((4\pi)/(3)\right))`

The angle 4π/3 is found in quadrant III of the unit circle, where sin(4π/3) = -√3/2 and cos(4π/3) = -1/2. Substituting these values gives:

`\begin{aligned}\cot\left((4\pi)/(3)\right) &= (-(1)/(2))/(-(√(3))/(2))\\\\&= ((1)/(2))/((√(3))/(2))\\\\&=(1)/(√(3))\\\\&=(1 \cdot √(3))/(√(3)\cdot √(3))\\\\&=(√(3))/(3)\end{aligned}`

So, the exact value of cot(4π/3) is:

`\large\boxed{\boxed{\cot\left((4\pi)/(3)\right)=(√(3))/(3)}}`