Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 10 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (10, 0) and then rotates through an angle of −225°, what is the new location of the hand?
(10,0) is on the x-axis, so if you rotate -225°, you are going clockwise 225° which is coterminal with 135°, in the 2nd quadrant
the coordinates would be (10cos135°, 10sin135‚
= (10(-cos45), 10 sin45°)
= (10(-√2/2) , 10(√2/2) )
= ( -5√2 , 5√2 )
(10,0) is 10 units from the origin along the x-axis
negative angles are clockwise rotations
... so the terminal angle is a 45º reference angle in Quad II
x = 10 cos(-225º)
y = 10 sin(-225º)
To find the new location of the robot's hand, we need to understand the concept of rotating a point in a coordinate plane.
When a point is rotated counterclockwise around the origin in a coordinate plane, we can use the following formulas to find the new coordinates:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
where (x, y) are the original coordinates, (x', y') are the new coordinates after the rotation, and θ is the angle of rotation.
In this case, the original coordinates of the robot's hand are (10, 0) and the angle of rotation is -225°.
First, we need to convert the angle from degrees to radians because the trigonometric functions in most programming languages work with radians. We can use the formula: radians = degrees * (π/180)
So, -225° in radians would be: -225 * (π/180) = -225π/180
Now, we can substitute the values into the rotation formulas:
x' = 10 * cos(-225π/180) - 0 * sin(-225π/180)
y' = 10 * sin(-225π/180) + 0 * cos(-225π/180)
After simplifying, we get:
x' = -7.07
y' = -7.07
So, the new location of the robot's hand after rotating through an angle of -225° is approximately (-7.07, -7.07).