Lesson 7.3 : Unit Circle

New Concept

The unit circle redefines trigonometric functions. By relating an angle $t$ to a point $(x, y)$ on a circle with radius 1, we define $\operatorname{cos} t = x$ and $\operatorname{sin} t = y$. This visual approach lets us find values for any angle, not just acute ones.

What’s next

This is just the starting point. Next, you'll use interactive examples to find sine and cosine for special angles and apply reference angles to solve problems.

Property

A unit circle has a center at $(0, 0)$, and radius 1. In a unit circle, the length of the intercepted arc is equal to the radian measure of the central angle $t$.

Let $(x, y)$ be the endpoint on the unit circle of an arc of arc length $s$. The $(x, y)$ coordinates of this point can be described as functions of the angle.

Examples

• The unit circle's equation is $x^2 + y^2 = 1$. If a point on the circle has an x-coordinate of $x = \frac{3}{5}$, then $(\frac{3}{5})^2 + y^2 = 1$, which gives $y^2 = \frac{16}{25}$, so $y = \pm\frac{4}{5}$.

Property

If $t$ is a real number and a point $(x, y)$ on the unit circle corresponds to a central angle $t$, then

To find the sine and cosine from a point $P(x, y)$ on the unit circle, simply identify the coordinates. The sine of the angle $t$ is its y-coordinate, and the cosine is its x-coordinate.

Examples

• A point $P$ on the unit circle corresponds to an angle $t$ and has coordinates $(\frac{8}{17}, -\frac{15}{17})$. From the definition, we have $\operatorname{cos} t = \frac{8}{17}$ and $\operatorname{sin} t = -\frac{15}{17}$.

• An angle of $\pi$ radians ($180^\circ$) corresponds to the point $(-1, 0)$ on the unit circle. Therefore, $\operatorname{cos}(\pi) = -1$ and $\operatorname{sin}(\pi) = 0$.

Property

The Pythagorean Identity states that, for any real number $t$,

To find the cosine given the sine and the quadrant, substitute the sine value into the identity, solve for $\operatorname{cos} t$, and choose the positive or negative solution based on the quadrant's sign for x-values.

Examples

• If $\operatorname{sin}(t) = \frac{12}{13}$ and $t$ is in quadrant II, find $\operatorname{cos}(t)$. Using the identity, $\operatorname{cos}^2 t + (\frac{12}{13})^2 = 1$, so $\operatorname{cos}^2 t = 1 - \frac{144}{169} = \frac{25}{169}$. In quadrant II, cosine is negative, so $\operatorname{cos}(t) = -\frac{5}{13}$.

• If $\operatorname{cos}(t) = -\frac{7}{25}$ and $t$ is in quadrant III, find $\operatorname{sin}(t)$. We have $(-\frac{7}{25})^2 + \operatorname{sin}^2 t = 1$, so $\operatorname{sin}^2 t = 1 - \frac{49}{625} = \frac{576}{625}$. In quadrant III, sine is negative, so $\operatorname{sin}(t) = -\frac{24}{25}$.

Property

For special angles in the first quadrant, the sine and cosine values are derived from 30-60-90 and 45-45-90 triangles within the unit circle.

• For $t = 30^\circ$ or $\frac{\pi}{6}$: $(\operatorname{cos} t, \operatorname{sin} t) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

• For $t = 45^\circ$ or $\frac{\pi}{4}$: $(\operatorname{cos} t, \operatorname{sin} t) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

Property

For the sine and cosine functions, the domain and range define the set of all possible inputs (angles) and outputs (values).

• Domain: The input for $f(t) = \operatorname{sin} t$ and $g(t) = \operatorname{cos} t$ can be any real number. Domain: $(-\infty, \infty)$.

• Range: The output values for both sine and cosine are bounded by the unit circle's radius. Range: $[-1, 1]$.

Property

A reference angle, $t'$, is the acute angle formed by the terminal side of an angle $t$ and the horizontal axis. It is always between $0$ and $90^\circ$ ($0$ and $\frac{\pi}{2}$ radians).

• Quadrant I: $t' = t$
• Quadrant II: $t' = \pi - t$ or $180^\circ - t$
• Quadrant III: $t' = t - \pi$ or $t - 180^\circ$
• Quadrant IV: $t' = 2\pi - t$ or $360^\circ - t$

The sine and cosine of an angle have the same absolute value as their reference angle; the sign (+ or -) depends on the quadrant.