The point (-8,3) lies on the terminal arm of angle theta in standard position. Determine the primary trigonometric ratios for angle theta and calculate the value of theta to the nearest degree.

Only the sine ratio was positive, and when I used that to calculate the angle, it worked, but the other ratios got different answers.

by Pythagoras you should have found r to be √73

so sinØ = 3/√73
cosØ = -8/√73
tanØ = -3/8

If you want to test your answers using your calculator you must consider the CAST rule

Calculators are programmed to give you the answer which is "closest" to the origin of rotation.
e.g. arctan (-3/8) gives us -20.556° which is coterminal with 360 - 20.556 or 339.44

so our angle is in II so it would be 180-20.556 or 159.444°
Check my answers with that angle, they are correct.

(-24,-7)

If only the sine ratio was positive, we can determine that the angle lies in the second quadrant. Let's calculate the primary trigonometric ratios based on this information.

We are given the point (-8,3), which lies on the terminal arm of angle θ. To determine the primary trigonometric ratios for angle θ, we need to calculate the values of sine (sin θ), cosine (cos θ), and tangent (tan θ).

Using the given point (-8,3), we can determine the values as follows:

sin θ = opposite/hypotenuse = 3/√(8² + 3²) = 3/√(64 + 9) = 3/√73

cos θ = adjacent/hypotenuse = -8/√(8² + 3²) = -8/√(64 + 9) = -8/√73

tan θ = opposite/adjacent = 3/(-8) = -3/8

So, the primary trigonometric ratios for this angle are:

sin θ = 3/√73
cos θ = -8/√73
tan θ = -3/8

Now, let's find the value of θ to the nearest degree. To do this, we can use the inverse sine function (sin⁻¹) since we know the value of sine (sin θ).

θ ≈ sin⁻¹(3/√73) ≈ 20°

Therefore, the approximate value of θ to the nearest degree is 20°.

It is important to note that when determining the primary trigonometric ratios, we need to be careful with the signs based on the quadrant in which the point lies. In this case, only the sine ratio was positive, indicating that the angle θ lies in the second quadrant.

To determine the primary trigonometric ratios for angle θ and calculate its value, we need to consider the position of the point (-8,3) on the coordinate plane.

We know that the point (-8,3) lies on the terminal arm of angle θ in standard position. In the coordinate plane, the x-coordinate (-8) represents the horizontal displacement and the y-coordinate (3) represents the vertical displacement from the origin.

Since the y-coordinate (3) is positive and no other ratio is positive, we can conclude that the only positive trigonometric ratio in this scenario is the sine ratio.

The sine ratio is defined as the opposite side (vertical displacement, in this case, 3) divided by the hypotenuse. To calculate the sine ratio, we need to determine the value of the hypotenuse.

Using the given coordinates, we can apply the Pythagorean theorem to find the length of the hypotenuse:

hypotenuse = sqrt((-8)^2 + 3^2) = sqrt(64 + 9) = sqrt(73)

The sine ratio is therefore:

sin θ = opposite/hypotenuse = 3/sqrt(73)

To calculate the value of angle θ to the nearest degree, we can use the inverse sine or arcsin function. Since the only positive ratio was the sine, we can use the inverse sine function directly.

θ ≈ arcsin(3/sqrt(73))

Calculating this using a scientific calculator, the angle θ is approximately 22 degrees (rounded to the nearest degree).

Therefore, the primary trigonometric ratios for angle θ are:

sin θ ≈ 0.391 (rounded to three decimal places)
cos θ ≈ -0.921 (rounded to three decimal places)
tan θ ≈ -0.424 (rounded to three decimal places)

It's important to note that since the point (-8,3) lies in the third quadrant, the cosine and tangent ratios would be negative. However, the sine ratio is positive because the y-coordinate is positive.