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### Course: Pixar in a Box > Unit 14

Lesson 2: Mathematics of rotation# 1. How do we rotate points?

First, let's think about how to rotate some really simple points such as (0,0) and (1,0). We use sine and cosine formulas to figure out where things end up after we spin them. Click here to review basic trig ratios..

## Want to join the conversation?

- Can someone explain this to me? I don't understand the exercise after this video. PLEASE!!!(10 votes)
- Well, this involves Trigonometry, so if you're not at that level yet (I'm not, so I barely understand myself) it might not make sense.(5 votes)

- I have a question. If a banana is red is it rotten?(5 votes)
- do we have to do this for h .w(2 votes)
- Wouldn't quaternions be applicable (and faster) here?(3 votes)
- Why does a positive rotation mean the object rotates counter-clockwise? Every software I've ever seen uses positive values for clockwise rotation.(2 votes)
- Really? Counterclockwise is the standard way to rotate objects with a positive angle: https://en.wikipedia.org/wiki/Rotation_matrix#In_two_dimensions(1 vote)

- em habin tuble undestandwing dis vidweo. em ownly fwive(2 votes)
- a bit hard with formulas(1 vote)
- For anyone who has trouble in consequent practice questions for angles >90 degrees: remember that when defined generally cosine function just gives x coordinate on the unit circle for given angle. And sine function is defined as y coorinate for given angle on unit circle. Trig function is defined as sine function divided by cosine function. There is no even need for any proof, this is how they are defined.(1 vote)
- im in 8th grade how would i know this(1 vote)

## Video transcript

(steps and bouncing) (switch clicks) - In this tutorial, we're
gonna take a closer look at the algebra of rotations. We'll be working with the
trigonometric functions known as sine and cosine. Let's start by rotating about
the origin by an angle theta. The usual convention is that rotating by a positive angle is a
counterclockwise motion, and rotating by a negative
angle is a clockwise motion. We'd like to find a formula that tells us where every
point x, y goes when rotated. Let's let x prime, y
prime be the coordinates of the point x, y after rotation. We wanna find formulas
for x prime and y prime, in terms of x, y, and theta. One such point is really easy. What happens to the point
zero, zero when rotated? It stays still. So x prime equals zero,
and y prime equals zero. What about the point one, zero? It gets rotated to a point x
prime, y prime as shown here. To determine formulas for x
prime and y prime in this case, drop a perpendicular from x
prime, y prime to the x axis. The orange length is x prime, and the magenta length is y prime. Notice that the orange, magenta, and green triangle is a right triangle. Notice that the length of the green line, the hypotenuse of the triangle, is one, because the point one,
zero is one unit away from the origin. And the lengths don't
change when you rotate. Notice too that the magenta
line is the line opposite theta, and it has length y prime,
which we don't know just yet. The ratio of the opposite side over the hypotenuse is sine theta. That is, y prime over
one equals sine theta, or in other words, y
prime equals sine theta. (switch clicks on) (gentle ringing) Similarly, the orange
line is adjacent to theta, and has length x prime, so if I form the ratio of
adjacent over hypotenuse, I get x prime over one
equals cosine theta, meaning that x prime equals cosine theta. This tells me that the
point one, zero gets rotated to the point cosine theta, sine theta. Use the next exercise to get
some practice with these ideas.