Grade 10

Advanced

9 min

Lesson 8 of 12

The Unit Circle and Trigonometric Identities

Not Started

Connects trigonometry to the coordinate plane and introduces fundamental identities.

What if a single circle with a radius of one could unlock the secrets of every wave in the universe—from the light hitting your eyes to the sound vibrating in your ears?

Learning Objectives

1.Define trigonometric functions as coordinates (x, y) on a circle with radius 1.
2.Prove and apply the fundamental Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$ .
3.Calculate exact trigonometric values for special angles like $30^\circ$ , $45^\circ$ , and $60^\circ$ without a calculator.

The Circle that Simplifies Everything

In basic geometry, trigonometry is about right triangles. But in advanced math, we move to the Unit Circle: a circle centered at the origin $(0,0)$ with a radius of exactly $1$ . When we draw an angle $\theta$ from the positive x-axis, it intersects the circle at a point $P(x, y)$ . Because the radius (hypotenuse) is $1$ , the definitions of sine and cosine simplify beautifully. Instead of 'opposite over hypotenuse,' we simply say: ** $\cos(\theta) = x$ and $\sin(\theta) = y$ **. This transition allows us to define trigonometry for any angle, even those greater than $90^\circ$ or less than $0^\circ$ .

Quick Check

If a point on the unit circle is located at $(0.6, 0.8)$ , what is the value of $\sin(\theta)$ ?

Remember that on the unit circle, the y-coordinate represents the sine of the angle.

Answer

0.8

The Pythagorean Identity

Since every point

(x, y)

on the unit circle forms a right triangle with the x-axis, we can apply the Pythagorean Theorem:

a^2 + b^2 = c^2

. In our circle, the legs are

x

and

y

, and the hypotenuse is the radius

1

. Substituting our trig definitions (

x = \cos\theta

and

y = \sin\theta

), we derive the most important identity in trigonometry:

\cos^2(\theta) + \sin^2(\theta) = 1

This formula is a mathematical 'law of conservation'—no matter what the angle is, the sum of the squares of sine and cosine will always equal one.

Suppose you know that $\cos(\theta) = \frac{3}{5}$ and the angle is in the first quadrant. Find $\sin(\theta)$ . 1. Start with the identity: $\sin^2(\theta) + \cos^2(\theta) = 1$ . 2. Substitute the known value: $\sin^2(\theta) + (\frac{3}{5})^2 = 1$ . 3. Square the fraction: $\sin^2(\theta) + \frac{9}{25} = 1$ . 4. Subtract from 1: $\sin^2(\theta) = \frac{16}{25}$ . 5. Take the square root: $\sin(\theta) = \frac{4}{5}$ .

Special Angles and Exact Values

While calculators give decimals, mathematicians prefer exact values. These come from two special triangles: the $45-45-90$ and the $30-60-90$ . For $45^\circ$ , the $x$ and $y$ coordinates are equal: $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ . For $30^\circ$ , the point is $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ , and for $60^\circ$ , it flips to $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ . Memorizing these coordinates allows you to solve complex physics and engineering problems with perfect precision.

Find the exact value of $\tan(60^\circ)$ using unit circle coordinates. 1. Recall that $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ , which is $\frac{y}{x}$ . 2. Identify coordinates for $60^\circ$ : $x = \frac{1}{2}$ , $y = \frac{\sqrt{3}}{2}$ . 3. Set up the fraction: $\tan(60^\circ) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}$ . 4. Simplify by multiplying by the reciprocal: $\frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3}$ .

Quick Check

What is the x-coordinate (cosine) of an angle of $90^\circ$ on the unit circle?

90^\circ

, the point is at the very top of the circle on the y-axis.

Answer

Find the exact value of $\cos(150^\circ)$ . 1. Locate $150^\circ$ on the circle. It is in Quadrant II ( $90^\circ$ to $180^\circ$ ). 2. Find the reference angle: $180^\circ - 150^\circ = 30^\circ$ . 3. Use the $30^\circ$ x-coordinate: $\frac{\sqrt{3}}{2}$ . 4. Determine the sign: In Quadrant II, x-values are negative. 5. Result: $\cos(150^\circ) = -\frac{\sqrt{3}}{2}$ .

Key Takeaways

1On the unit circle ( $r=1$ ), the coordinates of any point are $(\cos\theta, \sin\theta)$ .
2The Pythagorean Identity $\sin^2\theta + \cos^2\theta = 1$ is derived from the equation of a circle $x^2 + y^2 = 1$ .
3Tangent is the ratio of sine to cosine: $\tan\theta = \frac{y}{x}$ .
4Special angles ( $30^\circ, 45^\circ, 60^\circ$ ) provide exact radical values used throughout calculus and physics.

Quick Check

Multiple Choice

If $\sin(\theta) = -1$ , where is the point located on the unit circle?

Multiple Choice

What is the exact value of $\sin(45^\circ)$ ?

True/False

The equation $\sin^2(80^\circ) + \cos^2(80^\circ) = 1$ is true.

What's Next?

Review Tomorrow

Tomorrow morning, try to sketch the unit circle from memory and label the (x, y) coordinates for $0^\circ, 90^\circ, 180^\circ,$ and $270^\circ$ .

Practice Activity

Pick any random angle $\theta$ . Use your calculator to find $\sin\theta$ and $\cos\theta$ , then square both and add them together to see if you get 1!

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The Unit Circle and Trigonometric Identities

Learning Objectives

The Circle that Simplifies Everything

The Pythagorean Identity

Special Angles and Exact Values

Key Takeaways

Quick Check

What's Next?

The Unit Circle and Trigonometric Identities

Learning Objectives

The Circle that Simplifies Everything

The Pythagorean Identity

Special Angles and Exact Values

Key Takeaways

Quick Check

What's Next?