The point P(2, -5) is located on the terminal arm of an angle θ in standard position.

a) Draw and label an appropriate diagram. Remember to label θ and all the sides of the triangle.

b) Determine the exact values of the primary trigonometric ratios for angle θ.

c) Determine the measure of angle θ to the nearest degree.

a) Here is the diagram of the angle θ in standard position with point P(2, -5) labeled:

```
|
|
P |
|_______
θ
```

The point P(2, -5) is located on the terminal arm of θ, which means it is on the unit circle. We can form a right triangle by connecting the origin (0, 0) to point P (2, -5). The side opposite to angle θ is the y-value (-5) of point P, and the side adjacent to angle θ is the x-value (2) of point P.

b) To determine the primary trigonometric ratios for angle θ, we can use the values of the sides of the right triangle formed in the diagram:

- The hypotenuse (r) can be found using the Pythagorean theorem: r = sqrt(2^2 + (-5)^2) = sqrt(4+25) = sqrt(29).
- The opposite side (y) is -5.
- The adjacent side (x) is 2.

Using these values, we can calculate the ratios:

- sine (sinθ) = opposite/hypotenuse = -5/sqrt(29)
- cosine (cosθ) = adjacent/hypotenuse = 2/sqrt(29)
- tangent (tanθ) = opposite/adjacent = -5/2
- cosecant (cscθ) = 1/sinθ = sqrt(29)/-5
- secant (secθ) = 1/cosθ = sqrt(29)/2
- cotangent (cotθ) = 1/tanθ = -2/5

c) To determine the measure of angle θ to the nearest degree, we can use the inverse trigonometric functions. Since we know the values of the sine and cosine, we can use inverse sine (arcsin) or inverse cosine (arccos) to find the angle.

Using inverse sine (arcsin) and rounding to the nearest degree, we get:

θ ≈ arcsin(-5/sqrt(29)) ≈ -14°

Therefore, the measure of angle θ to the nearest degree is -14°.

a) To draw the appropriate diagram, we can plot the point P(2, -5) on the coordinate plane. The x-coordinate of P is 2, which means it is 2 units to the right of the origin, and the y-coordinate is -5, which means it is 5 units below the origin.

The angle θ is formed by the positive x-axis and the line connecting the origin to the point P. We can draw a right triangle with the hypotenuse being the line connecting the origin to P. The base of the triangle is the x-coordinate (2) and the height is the y-coordinate (-5).

The hypotenuse, which is the line connecting the origin to P, can be found using the Pythagorean theorem:
hypotenuse^2 = base^2 + height^2
hypotenuse^2 = 2^2 + (-5)^2
hypotenuse^2 = 4 + 25
hypotenuse^2 = 29
hypotenuse = sqrt(29)

Label the angle θ, the base (2), the height (-5), and the hypotenuse (sqrt(29)) on the diagram.

b) To determine the exact values of the primary trigonometric ratios for angle θ, we need to use the sides of the right triangle we formed in the previous step.

The primary trigonometric ratios are:

- Sine (sinθ) = opposite / hypotenuse = (-5) / sqrt(29)
- Cosine (cosθ) = adjacent / hypotenuse = 2 / sqrt(29)
- Tangent (tanθ) = opposite / adjacent = (-5) / 2

c) To determine the measure of angle θ to the nearest degree, we can use the inverse trigonometric functions.

- The measure of angle θ in degrees can be found using the inverse sine function (sin^(-1)). We can find sin^(-1)((-5) / sqrt(29)) using a scientific calculator or online tool, which gives approximately -47.99 degrees. Since the angle is in the third quadrant, we can add 180 degrees to get the positive measure: -47.99 + 180 = 132.01 degrees.

Therefore, the measure of angle θ to the nearest degree is 132 degrees.

Draw the angle. It should be clear that

y = -5
x = 2
r = √29

Now just recall that
sinθ = y/r
cosθ = x/r
tanθ = y/x

now let 'er rip

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