# Lesson 5

The Pythagorean Identity (Part 1)

- Let’s learn more about cosine and sine.

### 5.1: Circle Equations

Here is a circle centered at \((0,0)\) with a radius of 1 unit.

What are the exact coordinates of \(P\) if \(P\) is rotated counterclockwise \(\frac{\pi}{3}\) radians from the point \((1,0)\)? Explain or show your reasoning.

### 5.2: Cosine, Sine, and the Unit Circle

What are the exact coordinates of point \(Q\) if it is rotated \(\frac{2\pi}{3}\) radians counterclockwise from the point \((1,0)\)? Explain or show your reasoning.

### 5.3: A New Identity

- Is the point \(\left(\text-0.5,\sin(\frac{4\pi}{3})\right)\) on the unit circle? Explain or show your reasoning.
- Is the point \(\left(\text-0.5,\sin(\frac{5\pi}{6})\right)\) on the unit circle? Explain or show your reasoning.
- Suppose that \(\sin(\theta)=\text-0.5\) and that \(\theta\) is in quadrant 4. What is the exact value of \(\cos(\theta)\)? Explain or show your reasoning.

Show that if \(\theta\) is an angle between 0 and \(2\pi\) and neither \(\cos(\theta)=0\) nor \(\sin(\theta)=0\), then it is impossible for the sum of \(\cos(\theta)\) and \(\sin(\theta)\) to be equal to 1.

### Summary

Let’s say we have a point \(P\) with coordinates \((a,b)\) on the unit circle, like the one shown here:

Using the Pythagorean Theorem, we know that \(a^2+b^2=1\). We also know this is true using the equation for a circle with radius 1 unit, \(x^2+y^2=1^2\), which is true for the point \((a,b)\) since it is on the circle.

Another way to write the coordinates of \(P\) is using the angle \(\theta\), which gives the location of \(P\) on the unit circle relative to the point \((1,0)\). Thinking of \(P\) this way, its coordinates are \((\cos(\theta),\sin(\theta))\). Since \(a=\cos(\theta)\) and \(b=\sin(\theta)\), we can return to the Pythagorean Theorem and say that \(\cos^2(\theta) + \sin^2(\theta) = 1\) is also true.

What if \(\theta\) were a different angle and \(P\) wasn’t in quadrant 1? It turns out that no matter the quadrant, the coordinates of any point on the unit circle given by an angle \(\theta\) are \((\cos(\theta),\sin(\theta))\). In fact, the definitions of \(\cos(\theta)\) and \(\sin(\theta)\) are the \(x\)- and \(y\)-coordinates of the point on the unit circle \(\theta\) radians counterclockwise from \((1,0)\). Up until today, we’ve only been using the quadrant 1 values for cosine and sine to find side lengths of right triangles, which are always positive.

This revised definition of cosine and sine means that \(\cos^2(\theta) + \sin^2(\theta) = 1\) is true for all values of \(\theta\) defined on the unit circle and is known as the **Pythagorean Identity**.

### Glossary Entries

**Pythagorean identity**The identity \(\sin^2(x) + \cos^2(x) = 1\) relating the sine and cosine of a number. It is called the Pythagorean identity because it follows from the Pythagorean theorem.

**period**The length of an interval at which a periodic function repeats. A function \(f\) has a period, \(p\), if \(f(x+p) = f(x)\) for all inputs \(x\).

**periodic function**A function whose values repeat at regular intervals. If \(f\) is a periodic function then there is a number \(p\), called the period, so that \(f(x + p) = f(x)\) for all inputs \(x\).

**unit circle**The circle in the coordinate plane with radius 1 and center the origin.