# Lesson 5

The Pythagorean Identity (Part 1)

### 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

1. Is the point $$\left(\text-0.5,\sin(\frac{4\pi}{3})\right)$$ on the unit circle? Explain or show your reasoning.
2. Is the point $$\left(\text-0.5,\sin(\frac{5\pi}{6})\right)$$ on the unit circle? Explain or show your reasoning.
3. 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.