# How do you find the exact values of the six trigonometric function of #theta# if the terminal side of #theta# in the standard position contains the point (0,-6)?

Find 6 values of trig functions

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To find the exact values of the six trigonometric functions of theta ((\theta)), we can use the coordinates of the point where the terminal side of theta intersects the unit circle. Since the point is given as (0, -6), it lies on the negative y-axis, six units below the origin.

Given that the radius of the unit circle is 1, the hypotenuse of the triangle formed by the point (0, -6), the origin, and the x-axis is 6 units.

Using this information, we can calculate the values of the trigonometric functions:

- The sine function ((\sin)) is the y-coordinate divided by the hypotenuse, so (\sin(\theta) = \frac{-6}{6} = -1).
- The cosine function ((\cos)) is the x-coordinate divided by the hypotenuse. Since the x-coordinate is 0, (\cos(\theta) = \frac{0}{6} = 0).
- The tangent function ((\tan)) is the ratio of the sine to the cosine, so (\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-1}{0}). Division by zero is undefined, indicating that (\tan(\theta)) is undefined at this point.
- The cosecant function ((\csc)) is the reciprocal of the sine, so (\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{-1} = -1).
- The secant function ((\sec)) is the reciprocal of the cosine. Since (\cos(\theta) = 0), (\sec(\theta)) is undefined.
- The cotangent function ((\cot)) is the reciprocal of the tangent. Since (\tan(\theta)) is undefined, (\cot(\theta)) is also undefined.

Therefore, the exact values of the six trigonometric functions of theta ((\theta)) are as follows:

- (\sin(\theta) = -1)
- (\cos(\theta) = 0)
- (\tan(\theta)) is undefined
- (\csc(\theta) = -1)
- (\sec(\theta)) is undefined
- (\cot(\theta)) is undefined.

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