The point (r,theta) in polar coordinates is (7,5) in rectangular coordinates. What is the point (2r, theta + pi/2) in rectangular coordinates? So I tried this problem and got r = sqrt(74). But while finding the value of theta, I got it equal to cot(5/7) which ends up being some decimal number but I need it to be expressed as a square root or fraction in order to continue because I have to have my answer in exact form. Any help?
r = sqrt (74)
theta = tan^-1 (5/7)
leave it that way, do not go to degrees
2 r = 2 sqrt(74)
draw this axis system to figure out tan theta+pi/2
we are in Quadrant 2 theta degrees from the y axis
so now tan of new theta = 7/-5
and we have the point
( 2 sqrt 74 , tan^-1 (-7/5)
now from the drawing you can see that
x = -5 (2) = -10
and
y = 7 (2) = 14
so we are at
(-10,14)
For (r,θ),
x = r cosθ = 7
y = r sinθ = 5
For the new point, we have
x = 2r cos(θ+π/2) = -2r sinθ
y = 2r sin(θ+π/2) = 2r cosθ
so, rectangular coordinates for (2r,θ+π/2) are (-10,14)
To find the value of theta in exact form, you can use the inverse trigonometric functions. In this case, the value of theta can be found as follows:
tan(theta) = r2 / r1
tan(theta) = sqrt(74) / 7
Now, to find the value of theta, you can use the inverse tangent function to get:
theta = arctan(sqrt(74) / 7)
Since this value cannot be expressed in a more simplified exact form, you can leave it as theta = arctan(sqrt(74) / 7) to continue with the problem.
To find the point (2r, theta + pi/2) in rectangular coordinates, let's first calculate the value of r correctly.
We are given that the point (r, theta) in polar coordinates is (7, 5) in rectangular coordinates. From this, we can use the conversion formulas:
x = r * cos(theta)
y = r * sin(theta)
Plugging in the given values, we have:
x = 7 * cos(5)
y = 7 * sin(5)
Therefore, the rectangular coordinates of the point (7, 5) in polar coordinates are approximately (6.740, 0.612).
Now, let's calculate the values of x and y for the point (2r, theta + pi/2). We have:
x' = (2r) * cos(theta + pi/2)
y' = (2r) * sin(theta + pi/2)
Using the double angle identities for cosine and sine, we can rewrite these equations as:
x' = (2r) * cos(theta) * cos(pi/2) - (2r) * sin(theta) * sin(pi/2)
y' = (2r) * sin(theta) * cos(pi/2) + (2r) * cos(theta) * sin(pi/2)
Since cos(pi/2) = 0 and sin(pi/2) = 1, these simplify to:
x' = -2r * sin(theta)
y' = 2r * cos(theta)
Substituting the correct value of r into these equations, we get:
x' = -2 * sqrt(74) * sin(5)
y' = 2 * sqrt(74) * cos(5)
Therefore, the rectangular coordinates of the point (2r, theta + pi/2) are approximately (-5.716, 11.424).
Note: The value of theta you calculated, using cot(5/7) as approximated by a decimal, is not correct. The correct value is obtained by using the inverse tangent function:
theta = atan(5/7)
This yields theta approximately equal to 0.615 radians, which can't be expressed as a square root or fraction.