Sine, Cosine, and Tangent
Why this matters
Imagine you're designing a new video game. Your main character, Priya, is standing at the center of a large, circular plaza in a futuristic city like Seattle. A small security drone is programmed to fly in a perfect circle around her.
How do you, the programmer, keep track of the drone's exact location? You could track its distance east/west and its distance north/south from Priya. But what if the drone moves? Its coordinates are constantly changing.
What doesn't change is its circular path. A much smarter way to track it is by using the angle of rotation from a starting point (say, due east) and the radius of the circle.
This is exactly what we're going to do with trigonometry. We're going to see how an angle of rotation gives us a precise (x, y) coordinate, which in turn defines the core trigonometric functions: sine, cosine, and tangent.
Concept overview
flowchart TD
A[Start with an angle θ in standard position] --> B{Place on Unit Circle};
B --> C[Find intersection point P(x, y)];
C --> D[cos(θ) = x-coordinate];
C --> E[sin(θ) = y-coordinate];
C --> F{tan(θ) = slope};
F --> G[tan(θ) = y / x];
D --> G;
E --> G;
Core explanation
Welcome to one of the most foundational ideas in all of precalculus: the unit circle. It’s the tool that lets us unlock trigonometry for all angles, not just the acute angles inside a right triangle.
Angles in Standard Position
First, let's get our setup right. In the coordinate plane, we place an angle in standard position. This means two things:
- The angle's vertex is at the origin
(0, 0). - Its starting side, or initial ray, lies along the positive x-axis.
The other side of the angle, the terminal ray, is what rotates around the origin.
- A positive angle means you've rotated counterclockwise (like the hands on a clock going backward).
- A negative angle means you've rotated clockwise.
Imagine a spinning prize wheel at a carnival. The starting position is pointing to the right. Spinning it counterclockwise gives you a positive angle, while spinning it clockwise gives you a negative one. If you spin it all the way around (360° or 2π radians) and land in the same spot, you have a coterminal angle. For example, 30° and 390° are coterminal.
The Unit Circle: Our Measuring Tool
Now, let's introduce our main character: the unit circle. It's just a circle centered at the origin with a radius of exactly 1. That's it. The "unit" just means one.
Why is this so useful? Remember that the radian measure of an angle is the ratio of the arc length it cuts out to the radius (θ = s/r). On the unit circle, the radius r is 1. So, the formula becomes θ = s/1, or just θ = s.
This is a huge deal: On the unit circle, the radian measure of an angle is equal to the length of the arc it travels. An angle of π/2 radians literally means you've traveled a distance of π/2 along the circle's edge from the point (1, 0).
Defining Sine and Cosine as Coordinates
Okay, here's the heart of the lesson.
Take any angle θ in standard position. Its terminal ray will intersect the unit circle at some point, let's call it P. Since P is on the coordinate plane, it has coordinates (x, y).
Here is the big reveal:
- The cosine of the angle
θis defined as the x-coordinate of that point.cos(θ) = x - The sine of the angle
θis defined as the y-coordinate of that point.sin(θ) = y
So, the point P is literally (cos(θ), sin(θ)).
Think about it. For any angle you can imagine—positive, negative, huge, tiny—you just draw it in standard position, see where it hits the unit circle, and read the coordinates. The x-value is your cosine, and the y-value is your sine. It's that direct.
This is where some students get stuck. They try to think of a triangle first. Don't. Think of the coordinate first. Cosine is tied to the x-axis (horizontal displacement), and sine is tied to the y-axis (vertical displacement).
What about Tangent?
So if sine and cosine are the coordinates, what is tangent?
Remember that the slope of a line is "rise over run," or the change in y divided by the change in x. The terminal ray of our angle is a line segment starting at (0, 0) and going to (x, y).
The slope of this terminal ray is (y - 0) / (x - 0), which is just y / x.
And since we just defined y = sin(θ) and x = cos(θ), we get the third crucial definition:
- The tangent of the angle
θis the slope of the terminal ray.tan(θ) = y / x = sin(θ) / cos(θ)
This also tells us something important. What happens when x = 0? Division by zero is undefined. This happens at the top and bottom of the unit circle (at π/2 and 3π/2), where the terminal ray is perfectly vertical. That's why the tangent function has vertical asymptotes at those angles.
Let's recap the three main takeaways for the unit circle:
cos(θ) = xsin(θ) = ytan(θ) = y / x
This coordinate-based definition is more powerful than SOH CAH TOA because it works for any angle, allowing us to analyze repeating, cyclical patterns we see everywhere from sound waves to alternating currents.
Worked examples
Let's put these definitions into practice. It's one thing to see the formulas, but it's another to use them to get an answer.
Finding Trig Values for a Common Angle
Problem: The terminal side of an angle θ = π/4 intersects the unit circle at a point P. Determine sin(θ), cos(θ), and tan(θ).
Solution:
- 1Visualize the AngleAn angle of
π/4radians (which is 45°) is halfway through Quadrant I. We start at the positive x-axis and rotate counterclockwise. - 2Find the CoordinatesFor a
π/4(or 45°) angle, the intersection pointPon the unit circle is a well-known point. It forms a 45-45-90 right triangle with the x-axis. The coordinates for this point are(√2/2, √2/2). So,x = √2/2andy = √2/2. - 3Apply the DefinitionsNow we just map the coordinates to our trig functions.
cos(θ) = x: Therefore,cos(π/4) = √2/2.sin(θ) = y: Therefore,sin(π/4) = √2/2.tan(θ) = y / x: Therefore,tan(π/4) = (√2/2) / (√2/2). Anything divided by itself is 1. So,tan(π/4) = 1.
Working from a Point in a Different Quadrant
Problem: The terminal side of an angle t passes through the point P(-1/2, √3/2) on the unit circle. Find the values of sin(t), cos(t), and tan(t).
Solution:
- 1Identify the CoordinatesThe problem gives us the point of intersection directly. We don't even need to know the angle measure
tto find the trig values. We havex = -1/2andy = √3/2. - 2Check if it's on the Unit CircleA common check is to make sure the point is actually on the unit circle. The equation for the unit circle is
x² + y² = 1. Let's check:(-1/2)² + (√3/2)² = 1/4 + 3/4 = 4/4 = 1. Yes, it's on the unit circle. This step confirms we can use the simple definitions. - 3Apply the Definitions
cos(t) = x: So,cos(t) = -1/2.sin(t) = y: So,sin(t) = √3/2.tan(t) = y / x: So,tan(t) = (√3/2) / (-1/2).
- 4Simplify TangentThis is where students often make a small arithmetic mistake. To divide by
-1/2, we multiply by its reciprocal,-2/1.tan(t) = (√3/2) * (-2) = -√3.
Try it yourself
Ready to try one on your own? Don't worry about getting it perfect the first time; the goal is to practice the process.
Problem 1: An angle θ = 5π/6 is drawn in standard position.
- Which quadrant does the terminal ray lie in?
- The coordinates of the intersection point on the unit circle are
(-√3/2, 1/2). Use this to findcos(5π/6),sin(5π/6), andtan(5π/6).
Problem 2: The terminal ray of an angle α passes through the point (0, -1) on the unit circle. What are the values of sin(α), cos(α), and tan(α)?
Practice — 8 questions
In simple terms, this topic explains how an angle's sine, cosine, and tangent values are just the x and y coordinates of a point on a special circle with a radius of 1.
- 3.2.A: Determine the sine, cosine, and tangent of an angle using the unit circle.
- 3.2.A.1
- In the coordinate plane, an angle is in standard position when the vertex coincides with the origin and one ray coincides with the positive x-axis. The other ray is called the terminal ray. Positive and negative angle measures indicate rotations from the positive x-axis in the counterclockwise and clockwise direction, respectively. Angles in standard position that share a terminal ray differ by an integer number of revolutions.
- 3.2.A.2
- The radian measure of an angle in standard position is the ratio of the length of the arc of a circle centered at the origin subtended by the angle to the radius of that same circle. For a unit circle, which has radius 1, the radian measure is the same as the length of the subtended arc.
- 3.2.A.3
- Given an angle in standard position and a circle centered at the origin, there is a point, P, where the terminal ray intersects the circle. The sine of the angle is the ratio of the vertical displacement of P from the x-axis to the distance between the origin and point P. Therefore, for a unit circle, the sine of the angle is the y-coordinate of point P.
- 3.2.A.4
- Given an angle in standard position and a circle centered at the origin, there is a point, P, where the terminal ray intersects the circle. The cosine of the angle is the ratio of the horizontal displacement of P from the y-axis to the distance between the origin and point P. Therefore, for a unit circle, the cosine of the angle is the x-coordinate of point P.
- 3.2.A.5
- Given an angle in standard position, the tangent of the angle is the slope, if it exists, of the terminal ray. Because the slope of the terminal ray is the ratio of the vertical displacement to the horizontal displacement over any interval, the tangent of the angle is the ratio of the y-coordinate to the x-coordinate of the point at which the terminal ray intersects the unit circle; alternately, it is the ratio of the angle’s sine to its cosine.
flowchart TD
A[Start with an angle θ in standard position] --> B{Place on Unit Circle};
B --> C[Find intersection point P(x, y)];
C --> D[cos(θ) = x-coordinate];
C --> E[sin(θ) = y-coordinate];
C --> F{tan(θ) = slope};
F --> G[tan(θ) = y / x];
D --> G;
E --> G;
Read what Saavi narrates
(upbeat, warm intro music fades)
Hi everyone, it's Saavi. I'm so glad you're here.
Imagine you're designing a video game. Your character is standing at the center of a big, circular plaza. A security drone is flying in a perfect circle around her. How do you, the programmer, keep track of that drone? You could track its distance east and west, and its distance north and south... but a much smarter way is to use the angle of rotation.
That's exactly what we're going to do today. We're going to see how an angle gives us a precise x, y coordinate, which in turn defines sine, cosine, and tangent.
The big idea is that we're going to use something called the unit circle. It's a circle with a radius of one. And here's the secret: the x and y coordinates of a point on this circle *are* the cosine and sine of the angle that points to it.
Let's try an example. Say the terminal side of an angle, we'll call it t, passes through the point P, with coordinates negative one-half, and the square root of three over two... and this point is on the unit circle. We want to find the sine, cosine, and tangent of t.
Well, the problem gives us the point of intersection directly. The x-coordinate is negative one-half, and the y-coordinate is the square root of three over two.
The definition tells us that cosine of t is just the x-coordinate. So, cosine of t is negative one-half. Easy enough.
And sine of t is just the y-coordinate. So, sine of t is the square root of three over two.
Now for tangent. Tangent is the slope, which is y divided by x. So we have the square root of three over two... divided by negative one-half. Remember, when you divide by a fraction, you just multiply by the reciprocal. So we have the square root of three over two, times negative two. The twos cancel out, and we're left with negative square root of three.
Now, one of the most common mistakes I see every year is students mixing up sine and cosine. They'll write that sine is x and cosine is y. It's an easy mistake to make when you're in a hurry. The way to remember it is to think alphabetically. C for Cosine comes before S for Sine, just like x comes before y in an ordered pair. So the point is always (cosine, sine).
You're building a powerful new way of thinking about angles. Keep practicing, stay patient with yourself, and you will master this. You've got this.
(outro music fades in)
The definitions are tied to the axes. Sine is the vertical component (`y`), and cosine is the horizontal component (`x`).
Think alphabetically and spatially. `C` for Cosine comes before `S` for Sine, just like `x` comes before `y` in an ordered pair `(x, y)`. So it's `(cos θ, sin θ)`.
An angle of `2π/3` is in Quadrant II, where all `x` coordinates are negative. Since `cos(θ) = x`, the cosine value must be negative.
Before you calculate, always ask, "Which quadrant am I in?" and "Is `x` positive or negative there? Is `y` positive or negative there?" Use the mnemonic "All Students Take Calculus" (All, Sine, Tangent, Cosine) to remember which functions are positive in quadrants I, II, III, and IV, respectively.
Tangent is the slope of the terminal ray, which is "rise over run" or `Δy / Δx`. This corresponds to `y/x`.
Always remember `tan(θ) = sin(θ) / cos(θ)`. Since `sin(θ) = y` and `cos(θ) = x`, it must be `y/x`.
Angles can be negative or larger than 2π (one full revolution). The definitions work for any angle.
For an angle outside the `[0, 2π)` range, find its coterminal angle first by adding or subtracting multiples of 2π until you get an angle in that familiar range. For example, `cos(5π)` is the same as `cos(π)`, and `sin(-π/2)` is the same as `sin(3π/2)`.