Trigonometric Functions

Hey students! 👋 Welcome to one of the most fascinating topics in Algebra 2 - trigonometric functions! In this lesson, we'll explore how sine, cosine, and tangent work on the unit circle, learn to graph these amazing functions, and discover what amplitude and period mean. By the end of this lesson, you'll understand how these functions describe everything from sound waves to the motion of a Ferris wheel! 🎡

Understanding the Unit Circle and Basic Trigonometric Functions

Let's start with the foundation - the unit circle! 🔵 The unit circle is simply a circle with radius 1 centered at the origin (0,0) on a coordinate plane. This might seem basic, but it's actually one of the most powerful tools in mathematics!

When we place an angle θ (theta) in standard position - with its vertex at the origin and initial side along the positive x-axis - the terminal side of the angle intersects the unit circle at a specific point. This point has coordinates (x, y), and here's where the magic happens:

Cosine (cos θ) equals the x-coordinate of that point
Sine (sin θ) equals the y-coordinate of that point
Tangent (tan θ) equals sin θ divided by cos θ, or y/x

Think of it this way, students: imagine you're on a Ferris wheel with radius 1. As you rotate, your position can be described using these trigonometric functions! Your horizontal distance from the center is the cosine of your angle, and your vertical distance is the sine.

For example, at 90° (or π/2 radians), you're at the top of the circle at point (0, 1). So cos(90°) = 0 and sin(90°) = 1. At 0°, you're at (1, 0), giving us cos(0°) = 1 and sin(0°) = 0.

The tangent function is particularly interesting because it represents the slope of the line from the origin to your point on the circle. When you're at the top or bottom of the Ferris wheel (90° or 270°), the line is vertical, so tangent is undefined - this creates those dramatic vertical asymptotes we'll see in graphs! 📈

Graphing Sine and Cosine Functions

Now let's see what these functions look like when we graph them! 📊 The graphs of sine and cosine are called sinusoidal curves, and they're everywhere in nature - from ocean waves to your heartbeat on an EKG monitor.

The basic sine function y = sin(x) creates a smooth, wave-like curve that:

Starts at the origin (0, 0)
Reaches its maximum value of 1 at x = π/2 (90°)
Returns to 0 at x = π (180°)
Reaches its minimum value of -1 at x = 3π/2 (270°)
Completes one full cycle back at x = 2π (360°)

The cosine function y = cos(x) has the exact same shape, but it's shifted! It:

Starts at its maximum value of 1 when x = 0
Reaches 0 at x = π/2 (90°)
Hits its minimum of -1 at x = π (180°)
Returns to 0 at x = 3π/2 (270°)
Completes the cycle back at 1 when x = 2π (360°)

Here's a cool way to remember this, students: cosine is just sine shifted left by π/2 radians (90°). In fact, cos(x) = sin(x + π/2)!

Both functions oscillate between -1 and 1, creating those beautiful wave patterns. Real-world examples include:

Sound waves: The air pressure variations that create music follow sinusoidal patterns
AC electricity: The voltage in your home's electrical outlets follows a sine wave
Tides: Ocean levels rise and fall in patterns that can be modeled with these functions
Biorhythms: Many biological processes, like sleep cycles, follow sinusoidal patterns

Exploring the Tangent Function

The tangent function is the rebel of the trigonometric family! 😎 Unlike sine and cosine, which are well-behaved and stay between -1 and 1, tangent goes to infinity and beyond!

The graph of y = tan(x) has some unique characteristics:

It passes through the origin (0, 0)
It has vertical asymptotes (lines the function approaches but never touches) at x = π/2 + nπ, where n is any integer
Between each pair of asymptotes, the function increases from negative infinity to positive infinity
The function repeats this pattern every π radians (180°)

Think about why this happens, students! Remember that tan(x) = sin(x)/cos(x). When cosine equals zero (at 90°, 270°, etc.), we're dividing by zero, which creates those vertical asymptotes. It's like trying to calculate the slope of a perfectly vertical line!

Real-world applications of tangent include:

Architecture: Calculating the height of buildings using angles of elevation
Navigation: Determining distances and bearings
Engineering: Designing ramps and inclined planes

Understanding Amplitude and Period

Now let's dive into two crucial concepts that help us understand how trigonometric functions behave: amplitude and period! 📏

Amplitude is the maximum distance the function reaches from its middle (equilibrium) position. For the basic functions:

sin(x) and cos(x) both have an amplitude of 1
They oscillate between -1 and +1, so the amplitude is the distance from 0 to 1

But what if we have y = A·sin(x) where A is some number? The amplitude becomes |A|. For example:

y = 3sin(x) has an amplitude of 3 (oscillates between -3 and +3)
y = -2cos(x) has an amplitude of 2 (oscillates between -2 and +2)

Period is the horizontal distance required for the function to complete one full cycle and start repeating. For basic functions:

sin(x) and cos(x) have a period of 2π radians (360°)
tan(x) has a period of π radians (180°)

When we have y = sin(Bx) where B is a coefficient, the period becomes 2π/|B|. For instance:

y = sin(2x) has a period of 2π/2 = π (completes cycles twice as fast)
y = cos(x/3) has a period of 2π/(1/3) = 6π (takes three times as long to complete a cycle)

These concepts are incredibly useful in real life! Sound engineers use amplitude to control volume (louder sounds have greater amplitude) and frequency (related to period) to control pitch. Ocean scientists use these concepts to predict wave heights and timing. Even your smartphone's accelerometer uses trigonometric functions to detect motion! 📱

Conclusion

Congratulations, students! You've just mastered the fundamentals of trigonometric functions! 🎉 We explored how sine, cosine, and tangent emerge naturally from the unit circle, learned to visualize their distinctive graph patterns, and discovered how amplitude and period control their behavior. These functions aren't just abstract mathematical concepts - they're the mathematical language that describes waves, rotations, and oscillations throughout our world. From the music you listen to, to the GPS in your car, to the rhythm of your heartbeat, trigonometric functions are working behind the scenes to make sense of our dynamic universe!

Study Notes

• Unit Circle: Circle with radius 1 centered at origin; coordinates (cos θ, sin θ)

• Sine Function: y-coordinate on unit circle; y = sin(x)

Domain: all real numbers
Range: [-1, 1]
Period: 2π

• Cosine Function: x-coordinate on unit circle; y = cos(x)

Domain: all real numbers
Range: [-1, 1]
Period: 2π

• Tangent Function: tan(x) = sin(x)/cos(x)

Domain: all real numbers except π/2 + nπ
Range: all real numbers
Period: π
Vertical asymptotes at x = π/2 + nπ

• Amplitude: Maximum distance from equilibrium position

For y = A·sin(x) or y = A·cos(x), amplitude = |A|

• Period: Horizontal distance for one complete cycle

For y = sin(Bx) or y = cos(Bx), period = 2π/|B|
For y = tan(Bx), period = π/|B|

• Key Values:

sin(0) = 0, cos(0) = 1, tan(0) = 0
sin(π/2) = 1, cos(π/2) = 0, tan(π/2) = undefined
sin(π) = 0, cos(π) = -1, tan(π) = 0