The point A(-3, -5) is one the terminal arm of an angle feta. Determine the exact expressions for the primary trigonmetric ratios for the angle. Would that be 225 degrees.
-2V2?
(-3,-5) is in quadrant III
draw the triangle in that quad, the hypotenuse would be √34
sinØ = -5/√34 , csc Ø = =-√34/5
cosØ = -3/√34 , secØ = -√34/3
tanØ = 5/3, cotØ = 3/5
"feta" is not a Greek letter but it is a Greek cheese, however "theta" is a Greek letter.
If you listen carefully, your teacher more than likely is saying "Theta"
Ah, trigonometry, the art of confusing everyone with triangles. So, let's tackle this question together.
The point A(-3, -5) lies on the terminal arm of an angle, let's call it feta (pronounced like "fay-tah" not like the Greek cheese). Now, we want to determine the exact expressions for the primary trigonometric ratios.
First things first, we need to figure out the actual angle feta. And yes, you're right, it's actually 225 degrees (or 5π/4 radians if you want to get all mathy).
Now, let's find those trig ratios!
The primary trigonometric ratios are:
- Sine (sin)
- Cosine (cos)
- Tangent (tan)
To find them, we need to know the values of the coordinates of point A(-3, -5). The x-coordinate (-3) represents the adjacent side, and the y-coordinate (-5) represents the opposite side of our angle feta.
Now we can calculate the ratios:
Sine (sin) = Opposite / Hypotenuse
sin(feta) = -5 / Hypotenuse
Cosine (cos) = Adjacent / Hypotenuse
cos(feta) = -3 / Hypotenuse
Tangent (tan) = Opposite / Adjacent
tan(feta) = -5 / -3
Now, we're almost done! But we can simplify the expressions a bit.
Let's divide all the ratios by -3 (both numerator and denominator) to make it easier:
sin(feta) = -5 / (-3)
= 5/3
cos(feta) = -3 / (-3)
= 1
tan(feta) = -5 / (-3)
= 5/3
So, the exact expressions for the primary trigonometric ratios of the angle feta are:
sin(feta) = 5/3
cos(feta) = 1
tan(feta) = 5/3
And that's it! We did it! Trigonometry conquered! I hope my attempt to make this topic a bit more fun helped you out.
To determine the exact expressions for the primary trigonometric ratios (sine, cosine, and tangent) of an angle, we need to find the values of the sides of the triangle formed by the given point.
Let's start by plotting the point A(-3, -5) on the coordinate plane.
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A(-3,-5)
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First, let's find the length of the hypotenuse (r) by using the distance formula:
r = √((-3 - 0)² + (-5 - 0)²)
= √((-3)² + (-5)²)
= √(9 + 25)
= √34
Now, we can determine the values of the trigonometric ratios for this angle:
1. Sine (sin θ) = Opposite / Hypotenuse
Since the opposite side is -5 and the hypotenuse is √34, the sine of θ is -5 / √34.
2. Cosine (cos θ) = Adjacent / Hypotenuse
Since the adjacent side is -3 and the hypotenuse is √34, the cosine of θ is -3 / √34.
3. Tangent (tan θ) = Opposite / Adjacent
Since the opposite side is -5 and the adjacent side is -3, the tangent of θ is -5 / -3, which simplifies to 5 / 3.
Therefore, the exact expressions for the primary trigonometric ratios for the angle represented by point A(-3, -5) are:
sin θ = -5 / √34
cos θ = -3 / √34
tan θ = 5 / 3
Note: The angle itself is not 225 degrees until we specify which quadrant it lies in.
To determine the exact expressions for the primary trigonometric ratios of an angle, we need to find the values of the sine (sin), cosine (cos), and tangent (tan) functions for that angle.
Given that point A(-3, -5) lies on the terminal arm of the angle, we can find the values of sin, cos, and tan by using the coordinates of point A.
Step 1: Find the length of the hypotenuse of the right triangle formed by point A.
Using the Pythagorean theorem, we have:
(Length of hypotenuse)^2 = (-3)^2 + (-5)^2
(Length of hypotenuse)^2 = 9 + 25
(Length of hypotenuse)^2 = 34
Length of hypotenuse = sqrt(34)
Step 2: Determine the reference angle.
The reference angle is the acute angle between the terminal arm and the x-axis. Since point A is in the third quadrant, this angle will be in the second quadrant and its reference angle will be the angle formed when we draw a line from point A to the y-axis.
Using the coordinates of point A, we can calculate the reference angle as:
Reference angle = atan(-5 / -3) = atan(5/3)
Step 3: Calculate the trigonometric ratios.
Using the reference angle calculated above, we can determine the exact expressions for sin, cos, and tan.
sin(feta) = -sin(reference angle) = -sin(atan(5/3)) = -5/sqrt(34)
cos(feta) = cos(reference angle) = cos(atan(5/3)) = -3/sqrt(34)
tan(feta) = -tan(reference angle) = -tan(atan(5/3)) = 5/3
Therefore, the exact expressions for the primary trigonometric ratios of the angle are:
sin(feta) = -5/sqrt(34)
cos(feta) = -3/sqrt(34)
tan(feta) = 5/3
Note: The angle in the third quadrant is typically measured as a negative angle, so the angle expressed as 225 degrees would be -225 degrees.