The unit circle is a core concept in trigonometry. Think of it as a visual reference for understanding trig functions and how they relate to angles. The unit circle is simply a circle with a radius of 1, centered on a standard x-y coordinate plane.
In this guide, we’ll break down the different parts of a blank unit circle and how to use it to perform trig calculations.
We’ll cover angles, radians, trig functions, and how to fill out the unit circle yourself.
Understanding Angles and Radians on the Unit Circle
The unit circle represents angles from 0° all the way around to 360°. Certain key angles, like 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360°, are usually marked, because they correspond to specific points on the circle.
It’s important to know your quadrants! The unit circle is divided into four sections, and each one represents 90°.
The sign of the trigonometric functions changes based on the quadrant, so it’s important to understand which quadrant you’re working in.
Radians are another way to measure angles. In the unit circle, 2π radians is equal to 360°, and π (pi) is about 3.14.
It’s helpful to know how to convert between degrees and radians. Some common radian values are π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), π (180°), and 3π/2 (270°).
Trigonometric Functions: Sine, Cosine, and Tangent
The unit circle is a great way to visualize the relationships between angles and trigonometric functions. Here’s a quick rundown of sine, cosine, and tangent, as well as their reciprocal functions.
Sine (sin θ) and Cosine (cos θ)
On the unit circle, the cosine of an angle (cos θ) is the x-coordinate of the point where the terminal side of the angle intersects the circle. The sine of the angle (sin θ) is the y-coordinate of that point.
The values of sine and cosine always fall between -1 and 1, inclusive. Also, both sin θ and cos θ are periodic, repeating every 2π radians (or 360 degrees).
Tangent (tan θ)
The tangent of an angle (tan θ) is defined as sin θ / cos θ. You can also think of the tangent as the slope of the line that connects the origin to the point on the circle.
Tangent is undefined when cos θ = 0, which happens at 90° and 270° (π/2 and 3π/2 radians).
Reciprocal Trigonometric Functions
Cosecant (csc θ) is 1/sin θ, secant (sec θ) is 1/cos θ, and cotangent (cot θ) is 1/tan θ. You can easily use the unit circle to figure out the values of these reciprocal functions as well.
Reference Angles and Their Significance
Reference angles are the acute angles created between the terminal side of any angle and the x-axis. They are useful because they make it easier to calculate trigonometric values for angles that aren’t acute.
Here’s how to use them:
- Figure out the reference angle.
- Find the trigonometric value of the reference angle.
- Decide whether that value is positive or negative based on the quadrant of the original angle.
The most common reference angles are 30°, 45°, and 60°. These correspond to the ratios in special right triangles (30-60-90 and 45-45-90), making them easy to remember and use.
How to fill in a blank unit circle
If you’re trying to learn trigonometry or precalculus, filling in a blank unit circle is a great exercise. Here’s how to do it:
- First, mark the key angles in degrees and radians, starting with 0°, 30°, 45°, 60°, and 90°. Make sure you put them in the right quadrant!
- Then, figure out the (x, y) coordinates for each angle. Remember, (x, y) = (cos θ, sin θ). Special right triangles and reference angles will come in handy.
- Finally, remember that sine and cosine have different signs in each quadrant. The mnemonic “All Students Take Calculus” can help you remember which trig functions are positive in each quadrant.
Summary
The unit circle is an incredibly useful tool for understanding trigonometry and its many applications. By grasping the connections between angles, radians, and trigonometric functions on the unit circle, you can solve a bunch of different problems.
The best way to really learn the unit circle is to practice filling in blank unit circles. Then, when you’re working with trig functions and equations, use the unit circle as a quick and easy reference.