# The point #(-5,-2)# is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

Here's a diagram I made using Desmos:

The unknown angle

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To determine the exact values of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of the angle whose terminal side contains the point (-5, -2) in standard position, we first need to find the values of the adjacent side, opposite side, and hypotenuse of the right triangle formed by the point (-5, -2) and the origin (0, 0).

Using the distance formula, we find the hypotenuse: [ \text{Hypotenuse} = \sqrt{(-5 - 0)^2 + (-2 - 0)^2} = \sqrt{(-5)^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29} ]

The angle's adjacent side is the horizontal distance from the point (-5, -2) to the origin: [ \text{Adjacent side} = -5 ]

The angle's opposite side is the vertical distance from the point (-5, -2) to the origin: [ \text{Opposite side} = -2 ]

Now, we can determine the trigonometric functions: [ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{-2}{\sqrt{29}} ] [ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{-5}{\sqrt{29}} ] [ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{-2}{-5} = \frac{2}{5} ]

The reciprocal functions can be determined as follows: [ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\sqrt{29}}{-2} ] [ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\sqrt{29}}{-5} ] [ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{5}{2} ]

Therefore, the exact values of the six trigonometric functions of the angle whose terminal side contains the point (-5, -2) in standard position are as calculated above.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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