write the polar eqaution
3 = r cos (theta - 315)
in rectangular form.
3=r cos(a-315)
where theta=a
now 315=360-45=2pi-pi/4=-pi/4
therefore, -315=+pi/4
therefore,3=r cos(a-315)becomes
3=r cos(a+pi/4)
but cos(a+pi/4)= cosa*cospi/4 - sina*sinpi/4
3=r cos(a-315)
where theta=a
now 315=360-45=2pi-pi/4=-pi/4
therefore, -315=+pi/4
therefore,3=r cos(a-315)becomes
3=r cos(a+pi/4)
but cos(a+pi/4)= cosa*cospi/4 - sina*sinpi/4
= 1/root2(cosa-sina)
3= 1/root2(rcosa-rsina)
= 1/root2 (x-y)
3root2=x-y
y = x-3root2 is an equation of line
To convert the given polar equation into rectangular form (also known as Cartesian form), we need to use the following relationships:
x = r * cos(theta)
y = r * sin(theta)
Given the equation:
3 = r * cos(theta - 315)
To express this equation in rectangular form, we'll substitute the values of x and y into the equation:
x = r * cos(theta) = 3 * cos(theta - 315)
y = r * sin(theta)
Now, let's work with the x component first:
x = 3 * cos(theta - 315)
To simplify this expression, we'll use the angle addition formula for cosine:
cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)
Using this formula, we can rewrite the expression as:
x = 3 * (cos(theta) * cos(315) + sin(theta) * sin(315))
Since cos(315) is equal to cos(-45), and sin(315) is equal to sin(-45), we can further simplify the equation:
x = 3 * (cos(theta) * cos(-45) + sin(theta) * sin(-45))
Now, let's move on to the y component:
y = r * sin(theta)
There are no changes to be made for the y component since we do not have any angle adjustments.
Therefore, the rectangular form of the polar equation 3 = r * cos(theta - 315) is:
x = 3 * (cos(theta) * cos(-45) + sin(theta) * sin(-45))
y = r * sin(theta)