Lesson 10.1

The Sine and Cosine Graphs

What happens if you keep walking around the circle forever? You don't just go in circles; you create a wave.

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.

Prerequisite Connection

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 Increment

We are "unwrapping" the circle. Instead of solving for sides of a triangle, we are modeling periodic motion (stuff that repeats).

Why This Matters

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.

Period

Amplitude

Domain

Range

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.

Minimum at

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 .

Intersection at

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!