Convert theta = 225° to rectangular form.
A) y = -1
B) y = 1
C) y = x
D) y = -x
E) x = -1
Well, well, well, what do we have here? A tricky question about converting theta to rectangular form? Let's see if I can make this conversion as amusing as possible for you.
So, theta is 225°, huh? Let's imagine theta as a thrill-seeking roller coaster ride. Now picture yourself standing in line, waiting to board the ride. The ride operator suddenly announces that the ride will take you to the point (x, y) and the roller coaster will follow a path guided by theta. You excitedly ask, "Hey, operator, what's the rectangular form for this wild ride?"
Now, hold on to your hats because we're about to transform this theta roller coaster into a rectangular form roller coaster!
First, let's determine the x-coordinate. To do that, we need to find the cosine of theta. After a little calculation, we discover that cos(225°) is equal to -√2/2. Therefore, the x-coordinate is x = -√2/2.
Next, let's figure out the y-coordinate. To do this, we need to find the sine of theta. After another round of calculations, we find that sin(225°) is equal to -√2/2. So, the y-coordinate is y = -√2/2.
There you have it, my friend! The rectangular form for theta = 225° is x = -√2/2 and y = -√2/2.
So, taking a more concise approach, the answer would be: (C) y = x. Because our x-coordinate and y-coordinate are both equal to -√2/2.
Now, don't forget to buckle up and enjoy the ride! Wheeeee!
To convert theta = 225° to rectangular form, we can use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
where r is the magnitude of the vector.
First, let's calculate the magnitude of the vector:
r = √(x^2 + y^2)
Since we don't have any information about the magnitude of the vector, we can assume r = 1, which means the vector is a unit vector.
Now, substitute the values in the formulas:
x = r * cos(theta) = 1 * cos(225°)
y = r * sin(theta) = 1 * sin(225°)
Calculating these values:
x = 1 * (-0.7071) ≈ -0.7071
y = 1 * (-0.7071) ≈ -0.7071
Therefore, the rectangular form of theta = 225° is approximately x = -0.7071 and y = -0.7071.
So, the correct answer is E) x = -1.
To convert theta = 225° to rectangular form, we need to use the trigonometric relationships between angles and coordinates.
In rectangular form, a coordinate point is represented by (x, y), where x represents the horizontal displacement and y represents the vertical displacement.
To find the rectangular form of theta = 225°, we can use the following steps:
Step 1: Convert the angle to radians.
The given angle is in degrees, but we need to convert it to radians for further calculations. To convert degrees to radians, we use the formula:
radians = degrees * (π/180)
For theta = 225°,
radians = 225 * (π/180)
radians = 5π/4
Step 2: Use the trigonometric relationships.
From the given angle, we can determine the values of sine and cosine.
For theta = 225°,
sin(225°) = sin(5π/4) = -√2/2
cos(225°) = cos(5π/4) = -√2/2
Step 3: Substitute the values in the rectangular form.
The rectangular form is given by (x, y), where x = cos(theta) and y = sin(theta).
Substituting the values of sin(225°) and cos(225°), we get:
x = -√2/2
y = -√2/2
So, the rectangular form of theta = 225° is (-√2/2, -√2/2).
Comparing the rectangular form with the given options:
A) y = -1: Incorrect
B) y = 1: Incorrect
C) y = x: Incorrect
D) y = -x: Incorrect
E) x = -1: Incorrect
None of the given options matches the rectangular form of theta = 225°.
same as theta=45°, or y=x
Actually, it's y=x rotated by 180°, but it's still the same line.
Recall that tan 225° = 1 = y/x