Radians, Degrees, and a Calculator
Applications of Radian Measure
Length of Arc
Area of a Sector
Length of a Chord
Also, when applying the chord-length formula, it isn’t actually necessary for the angle measure to be in radians (as it is with the arc-length and area formulas), because the sine of the angle is the same whether the angle is in radians or degrees. However, it is still useful to have students practice converting from degrees to radians.
Circular Functions of Real Numbers
Students should notice that the height of the point tracing out the sine graph at any given stage is exactly the same (in graph-units) as the height of the point moving around the circle. (This may not be immediately obvious because the sine graph and the circle graph are depicted on slightly different scales.)
It may seem strange that the tangent line can get infinitely long when the sine and cosine lines can’t. Remind students that the length of the tangent line represents the ratio between the sine and cosine, and so it gets infinitely big as the cosine gets infinitesimally small.
Linear and Angular Velocity
You may need to stress that s represents distance and does not stand for “speed.” Students will still be prone to forget this when plugging in values without thinking too hard, so they may need reminding when they slip up.
Graphing Sine and Cosine Functions
You may need to stress that the amplitude is the greatest distance the wave gets from the center of the wave, so it is only half the distance between the minimum and maximum values.
Period and Frequency
Transformations of Sine and Cosine Graphs: Dilations
Translating Sine and Cosine Functions
Horizontal Translations (Phase Shift)
General Sinusoidal Graphs
Drawing Sketches/Identifying Transformations from the Equation
Examples 1 and 2 demonstrate two different ways to approach the problem of sketching a graph: starting with the horizontal and vertical translations, or starting with the amplitude and frequency. Students will probably find they prefer one method or the other, and there’s certainly no need to be strict about which one to use.
Also, some students may find it easier to sketch a complete curve at each step of the process until they end up with the final curve, while others may find it easier to simply sketch the key points of the graph, move them around as necessary, and not connect them with a curve until the final step. Again, either method should work fine; it may in fact be a good idea to point out that both methods exist.
Writing the Equation from a Sketch
Problems 6-10 should contain the instruction “Write an equation that describes the given graph.” Of course, either sine or cosine may be used.