Lab
Unit Circle
Drag P around the circle, move the angle slider, or type an angle such as 2pi/3. The horizontal leg of the triangle is the x-coordinate of P and the vertical leg is its y-coordinate, each with its sign. Before you move P into a new quadrant, say which of cos θ, sin θ and tan θ will be negative there, then check. The graph underneath traces y = sin θ or y = cos θ from 0 up to the angle you have reached.
Challenges
- Before you set θ to 5π/6, say which quadrant its terminal ray is in and the signs of cos θ and sin θ. Then set it and read the exact coordinates of P.
- Type −π/4. Which angle between 0 and 2π has the same terminal ray? Press Add 2π and check that P does not move.
- Find the two angles between 0 and 2π where tan θ is not defined. What are the coordinates of P at each, and why does the ray have no slope there?
- Set r to 3 and θ to 2π/3. What are the coordinates of P? Compare them with the coordinates when r is 1.
- With the sine graph showing, move θ from 0 to 2π. On which interval is sin θ decreasing? Switch to the cosine graph and answer the same question for cos θ.
- Find an angle between 0 and 2π where sin θ and cos θ are equal and both negative. What is tan θ there?