Introduction
In the last chapter, we trapped trigonometry inside a unit circle. But in the real world, things change over time. If we plot the value of as increases, we don't get a circle—we get a Wave.
Past Knowledge
Remember that is just the y-coordinate on the unit circle. We are simply graphing how that height changes as the angle spins.
Today's Goal
We are "unwrapping" the circle. Instead of solving for sides of a triangle, we are modeling periodic motion (stuff that repeats).
Future Success
This is the birth of the Derivative in Calculus. You will eventually prove that the "slope" of the sine wave at any point is exactly equal to the cosine value at that point.
Key Concepts
The Sine Graph (" The Wave")
The graph of starts at 0, goes up to 1, down to -1, and back to 0. This cycle repeats every radians.
The Cosine Graph ("The Bucket")
The graph of starts at the maximum of 1, goes down to -1, and comes back up. It looks like a bucket.
Notice that the Cosine graph is exactly the same shape as Sine, just shifted to the left by . They are "out of phase".
Worked Examples
Example 1: Identification
Identify whether the function starts at 0 or 5.
Check the Parent Function
The parent function is .
Evaluate at x=0
.
So .
The graph starts at 0 (the origin).
Example 2: Key Points
Find the coordinates of the minimum of on the interval .
Recall the Shape
Cosine starts High (1), goes to Zero, goes Low (-1), goes Zero, ends High (1).
Locate the "Low"
The lowest point happens halfway through the cycle, at radians.
Example 3: Intersection
Where do and cross each other in the first quadrant?
Think Unit Circle
We need the angle where the x-coordinate equals the y-coordinate.
Match the Values
At , sine is small.
At , cosine is small.
At (), both are .
Common Pitfalls
Confusing the Starts
"Sine starts at 0, Cosine starts at 1."
Many students get this backwards. Remember: because the x-coordinate is 1 at 0 degrees.
Real-World Application
Alternating Current (AC)
The electricity coming out of your wall socket is a sine wave. It oscillates between positive and negative voltage 60 times a second (60 Hz). If it didn't alternate, it wouldn't be able to travel long distances without losing power!
Practice Quiz
Practice Quiz
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