Graphs of the Sine and Cosine Functions

 Imagine the terminal side of an angle $\theta$ in standard position rotating smoothly around the Unit Circle. As $\theta$ changes, so do the values of $sin(\theta)$ (the y-value on the circle) and $cos(\theta)$ (the x-value). The values of these two functions change from 0 to 1 back to 0 to -1 and back to zero as the angle rotates around the Unit Circle. If we plot the values of the functions as $\theta$ changes, we see the graphs of the sine and cosine functions. Check or clear the checkboxes to show or hide either or both function graphs. You can drag the $\theta$ slider manually, or you can click the "PLAY" button at the lower left of the app to rotate $\theta$ automatically. At each value of $\theta$, the angle's terminal side intersects the unit circle at some (x, y) coordinate. These coordinates correspond to $x=cos(\theta)$ and $y=sin(\theta)$, as illustrated by the point on the graphs at the right. Make the connection between y (red) on the unit circle and $sin(\theta)$ (red) on the graph by hiding the cosine graph, then switch. Observe especially what happens on the graph when $\theta$ is on the x- or y-axis.

Cofunction Identities

 The CO function identities relate sine to COsine, tangent to COtangent, and secant to COsecant by COmplemetary angles. Click the "PLAY" button to see that the two acute angles are always complements.

Law of Sines - Ambiguous Case

 The Law of Sines is a formula that can be used to solve all SAA and ASA triangles. It can also be used for SSA triangles, but the triangle resulting from defining angle A and sides a and b depends on the length of side a. If a is too short (a < h), it does not reach the third side c, and no triangle is formed. If a is longer than b, a single triangle results. And if a is between h and b, side a can reach side c in two ways, resulting in two possible triangles. The lengths of a and b can be changed in the diagram. Length b can be changed by moving point X, and length a can be changed by adjusting the slider. When a < h, no triangle is formed. When a is between h and b, two possible triangles (shown in red and green) can be formed. When a > b, only one triangle is possible (shown all red).

Conic Sections

[b]Check Out Conic Sections![/b][br][br][list][br][*]Check or Clear "Show Axes" to set visibility of x, y, and z axes[br][/*][*]Adjust Cutting Plane using Tilt (angle), Shift (left/right), and Height (up/down) sliders[br][/*][*]"Reset View" to restore default viewpoint and stop motion[br][/*][*]"Reset Plane" to restore cutting plane to starting orientation[br][/*][*]"Tilt Plane to Slant of Cone" sets plane tangent to cone edge (then adjust Shift or Height for true parabola)[br][/*][*]"View Above Section" sets view directly above (perpendicular to) section[br][/*][*]"Spin It!" to automatically revolve view around the z-axis[br][/*][*]"Stop" to stop revolving[br][/*][*]"Projection Type" button (4th) in 3D view to select 3D glasses[/*][/list][br][br]You can also manipulate the 3D view by [b]Right-Click-Dragging[/b]

Parametch-A-Sketch

 Move the sliders to turn the knobs manually. Change the equation in the box next to each knob to have the knobs follow them when you press START. (You should set the sliders to zero when using the automatic drawing feature). Screen scale is restricted to 0-6 for y and 0-9 for x.