Convert theta= 225° to rectangular form.
To convert an angle given in degrees, theta, to rectangular form, we can use the trigonometric functions - specifically, sine and cosine. The rectangular form consists of an x-coordinate (real part) and a y-coordinate (imaginary part), which represent a point on the complex plane.
In this case, the given angle is theta = 225°. To start, we need to find the cosine and sine of this angle.
1. Convert degrees to radians:
The trigonometric functions in most programming languages work with radian measurements, so we need to convert theta from degrees to radians.
theta_radians = theta * π / 180° [where π is approximately 3.14159]
Let's calculate it for theta = 225°:
theta_radians = 225° * π / 180° = 225 * π / 180
Simplifying further using the value of π:
theta_radians = 5π / 4
2. Find the cosine and sine of the angle:
cosine(theta_radians) = cos(5π / 4)
= cos(π + π/4) [since cos(π) = -1]
= cos(π) * cos(π/4) - sin(π) * sin(π/4)
= -1 * (1/√2) - 0 * (1/√2) [since cos(π) = -1 and sin(π) = 0]
= -1/√2
= -√2/2 [rationalizing the denominator]
sine(theta_radians) = sin(5π / 4)
= sin(π + π/4) [since sin(π) = 0]
= sin(π) * cos(π/4) + cos(π) * sin(π/4)
= 0 * (1/√2) + (-1) * (1/√2) [since sin(π) = 0 and cos(π) = -1]
= -1/√2
= -√2/2 [rationalizing the denominator]
3. Represent the rectangular form:
The rectangular form of a complex number is a + bi, where a is the real part (x-coordinate) and b is the imaginary part (y-coordinate).
In this case, we have:
a = cosine(theta_radians) = -√2/2
b = sine(theta_radians) = -√2/2
Therefore, the rectangular form of theta = 225° is:
-√2/2 - (√2/2)i