Question

Rewrite tan(3π/2) in terms of sines and cosines. sin(3π/2) / cos(3π/2)

Answer 1

tan(3π/2) is undefined because the cosine of 3π/2 is 0. Therefore, there is no valid solution for tan(3π/2) in terms of sines and cosines.

Step-by-step explanation:

To rewrite tan(3π/2) in terms of sines and cosines, we first need to find the values of sin(3π/2) and cos(3π/2). The sine of 3π/2 is -1 and the cosine of 3π/2 is 0. Therefore, tan(3π/2) = sin(3π/2) / cos(3π/2) = -1 / 0.

However, it's important to note that tan(3π/2) is undefined because the cosine of 3π/2 is 0. In trigonometry, dividing by zero is undefined. So, there is no valid solution for tan(3π/2) in terms of sines and cosines.

Answer 2

The tangent of
`\( (3\pi)/(2) \)`n by zero in the tangent expression. Therefore,
`( \tan\left((3\pi)/(2)\right) = (\sin\left((3\pi)/(2)\right))/(\cos\left((3\pi)/(2)\right)) \)` is not a valid mathematical operation.

Step-by-step explanation:

The tangent of an angle is defined as the ratio of the sine to the cosine of that angle. In the case of
`\(\tan\left((3\pi)/(2)\right)\),` we can use the values of sine and cosine for the angle
`\((3\pi)/(2)\).`

Starting with the expression
`\(\tan\left((3\pi)/(2)\right)\)`, we substitute the values of sine and cosine for
`\((3\pi)/(2)\)`:

`\[ \tan\left((3\pi)/(2)\right) = (\sin\left((3\pi)/(2)\right))/(\cos\left((3\pi)/(2)\right)) \]`

Now, we know that
`\(\sin\left((3\pi)/(2)\right) = -1\) and \(\cos\left((3\pi)/(2)\right) = 0\).`Dividing by zero is undefined in mathematics, so we should interpret this result accordingly. The tangent of
`\((3\pi)/(2)\)`is undefined, reflecting that the cosine is zero at this angle.

In conclusion, the expression
`\(\tan\left((3\pi)/(2)\right) = (\sin\left((3\pi)/(2)\right))/(\cos\left((3\pi)/(2)\right))\)`) is technically correct, but the result is undefined due to division by zero. This corresponds to the geometric understanding that the tangent function becomes infinite at angles where the cosine is zero.