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Sine and Cosine Function Graphs

Lesson ~10 min read 8 MCQs

In simple terms: In simple terms, graphing sine and cosine is about unwrapping the unit circle to visually track a point's vertical (sine) and horizontal (cosine) position as the angle changes.

Why this matters

Have you ever been on a big Ferris wheel, like the one at the Santa Monica Pier or Chicago's Navy Pier? As your car goes around, two things are constantly changing: your height above the ground and your horizontal distance from the center. At the very top, you're at your maximum height. At the bottom, you're at your lowest. When you're level with the center, your height is zero relative to that center point.

The graphs of sine and cosine functions work exactly like this. They are simply a way to plot that changing height (sine) or horizontal position (cosine) over time as you travel around a circle. Today, we're going to see how the familiar unit circle "unwraps" to create these beautiful, wavy graphs that show up everywhere from sound engineering to electrical grids.

The sine wave illustrates how vertical position changes as a point moves around a circle.

Concept overview

flowchart TD
    A[Start with an angle θ] --> B{Choose function};
    B --> C[y = sin(θ)];
    B --> D[y = cos(θ)];
    C --> E[Find point P(x, y) on unit circle];
    D --> E;
    E --> F{Get coordinate};
    F -- y-coordinate --> G[Value is y];
    F -- x-coordinate --> H[Value is x];
    G --> I[Plot (θ, y)];
    H --> J[Plot (θ, x)];
    I --> K[Repeat for next angle];
    J --> K;

Core explanation

Alright, let's connect the dots between the unit circle you already know and the graphs we're about to build. This is one of the most foundational ideas in trigonometry, so let's take our time with it.

From Unit Circle Coordinates to Function Values

Remember the unit circle? It's a circle with a radius of 1, centered at the origin (0,0). For any angle θ in standard position, the terminal ray intersects the circle at a point P(x, y).

We defined our core trig functions using this point:

  • The cosine of the angle, cos(θ), is the x-coordinate.
  • The sine of the angle, sin(θ), is the y-coordinate.

So, P(x, y) is the same as P(cos θ, sin θ).

This is the key. The sine and cosine functions simply take an angle θ as their input and output a coordinate value.

  • f(θ) = sin(θ) tracks the vertical position (the y-coordinate).
  • f(θ) = cos(θ) tracks the horizontal position (the x-coordinate).

The domain for both functions is all real numbers, because you can keep rotating around the circle infinitely in either the positive or negative direction.

Building the Sine Graph: y = sin(θ)

Let's build the sine graph piece by piece. On our graph, the horizontal axis will represent the angle θ (in radians), and the vertical axis will be the output of the sine function, sin(θ).

Imagine a point, let's call her Priya, starting at (1, 0) on the unit circle, which corresponds to an angle of θ = 0.

  1. At θ = 0: Priya is at (1, 0). The y-coordinate is 0. So, sin(0) = 0. We plot our first point on the graph at (0, 0).
  2. At θ = π/2: Priya travels a quarter of the way around the circle to (0, 1). Her height is now at its maximum. The y-coordinate is 1. So, sin(π/2) = 1. We plot the peak of our wave at (π/2, 1).
  3. At θ = π: She travels to the far left of the circle, at (-1, 0). Her height is back to the middle. The y-coordinate is 0. So, sin(π) = 0. We plot the x-intercept at (π, 0).
  4. At θ = 3π/2: She's at the very bottom of the circle, at (0, -1). Her height is at its minimum. The y-coordinate is -1. So, sin(3π/2) = -1. We plot the trough of our wave at (3π/2, -1).
  5. At θ = 2π: She completes one full revolution and is back where she started at (1, 0). The y-coordinate is 0 again. So, sin(2π) = 0. We plot the point (2π, 0).

If you connect these five key points with a smooth curve, you get one complete cycle of the sine wave. Notice how the output values oscillate, or wave, between -1 and 1. This pattern repeats every 2π radians because you're just going around the circle again.

Building the Cosine Graph: y = cos(θ)

Now, let's do the same thing for cosine. This time, we're tracking Priya's horizontal position, the x-coordinate.

  1. At θ = 0: Priya is at (1, 0). The x-coordinate is 1. So, cos(0) = 1. The cosine graph starts at its maximum, so we plot (0, 1).
  2. At θ = π/2: She's at (0, 1). Her horizontal position is now right in the middle. The x-coordinate is 0. So, cos(π/2) = 0. We plot the x-intercept at (π/2, 0).
  3. At θ = π: She's at (-1, 0). She is as far left as she can go. The x-coordinate is -1. So, cos(π) = -1. We plot the minimum point at (π, -1).
  4. At θ = 3π/2: She's at (0, -1). Her horizontal position is back to the middle again. The x-coordinate is 0. So, cos(3π/2) = 0. We plot another x-intercept at (3π/2, 0).
  5. At θ = 2π: She's back at (1, 0). The x-coordinate is 1 again. So, cos(2π) = 1. We plot the point (2π, 1).

Connecting these dots gives you one cycle of the cosine wave. It also oscillates between -1 and 1 and repeats every 2π.

  • Sine
    is the y-coordinate. At θ = 0, the y-coordinate is 0. The graph must start at (0,0).
  • Cosine
    is the x-coordinate. At θ = 0, the x-coordinate is 1. The graph must start at (0,1).

If you can ground yourself in that simple fact, you can always reconstruct these graphs from scratch.

Key points for sine and cosine functions derived from the unit circle.

Worked examples

Let's walk through how to apply this on paper. The key is to be systematic.

Example 1: Graphing y = sin(θ)

Problem: Identify the five key points for one cycle of the parent function f(θ) = sin(θ) starting at θ = 0, and sketch the graph.

Solution:

Our goal is to find the value of sin(θ) at the quadrantal angles: 0, π/2, π, 3π/2, and 2π. We'll do this by finding the y-coordinate of the point on the unit circle at each of those angles.

  1. Set up a table: This is the best way to keep your work organized.
θ (angle) Point on Unit Circle (cos θ, sin θ) y = sin(θ) (y-coordinate) Plotted Point (θ, y)
0 (1, 0) 0 (0, 0)
π/2 (0, 1) 1 (π/2, 1)
π (-1, 0) 0 (π, 0)
3π/2 (0, -1) -1 (3π/2, -1)
2π (1, 0) 0 (2π, 0)
  1. 1
    Explain the steps
    • For θ = 0, the point on the unit circle is (1, 0). The sine is the y-coordinate, which is 0.
    • For θ = π/2, the point is (0, 1). The sine is 1. This is our maximum value.
    • For θ = π, the point is (-1, 0). The sine is 0. We're back on the horizontal axis.
    • For θ = 3π/2, the point is (0, -1). The sine is -1. This is our minimum value.
    • For θ = 2π, we've completed a full circle and are back at (1, 0). The sine is 0.
  2. 2
    Sketch the graph
    • Draw your axes. Label the horizontal axis θ and mark 0, π/2, π, 3π/2, 2π.
    • Label the vertical axis y and mark 1 and -1.
    • Plot the five points from your table.
    • Connect them with a smooth, continuous wave.

Example 2: Graphing y = cos(θ)

Problem: Identify the five key points for one cycle of the parent function f(θ) = cos(θ) starting at θ = 0, and sketch the graph.

Solution:

The process is identical, but this time we focus on the x-coordinate.

  1. Set up the table:
θ (angle) Point on Unit Circle (cos θ, sin θ) y = cos(θ) (x-coordinate) Plotted Point (θ, y)
0 (1, 0) 1 (0, 1)
π/2 (0, 1) 0 (π/2, 0)
π (-1, 0) -1 (π, -1)
3π/2 (0, -1) 0 (3π/2, 0)
2π (1, 0) 1 (2π, 1)
  1. 1
    Explain the steps
    • For θ = 0, the point is (1, 0). The cosine is the x-coordinate, which is 1. The graph starts at its maximum.
    • For θ = π/2, the point is (0, 1). The cosine is 0.
    • For θ = π, the point is (-1, 0). The cosine is -1, our minimum value.
    • For θ = 3π/2, the point is (0, -1). The cosine is 0.
    • For θ = 2π, we're back at (1, 0). The cosine is 1.
  2. 2
    Sketch the graph
    • Draw and label your axes just like before.
    • Plot the five points: (0, 1), (π/2, 0), (π, -1), (3π/2, 0), and (2π, 1).
    • Connect them with a smooth, rounded curve. Notice it looks like a valley.

By building these graphs from the unit circle, you're not just memorizing a picture; you're understanding the very nature of these functions.

The sine function graph showing one full cycle from 0 to 2π.

Try it yourself

Ready to try on your own? Don't just jump to the answer; think through the connection to the unit circle.

  1. 1
    Problem
    Consider the angle θ = -π/2.
    • What are the coordinates of the point on the unit circle?
    • Using those coordinates, what are the values of sin(-π/2) and cos(-π/2)?
    • Where would these points be on their respective graphs?
  2. 2
    Problem
    Describe the shape of the y = cos(θ) graph over the interval from θ = -π to θ = π. How many full or partial cycles do you see?
The cosine function graph over the interval from -π to π.
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