The Unit Circle and Trigonometric Functions: A Precalculus Study Guide

Unit Circle Fundamentals

Definition and Equation of the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0, 0) in the coordinate plane. It is fundamental in trigonometry for relating angles to coordinates and trigonometric functions.

• Standard Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$ where (h, k) is the center and r is the radius.

• Unit Circle Equation: $x^2 + y^2 = 1$ (centered at (0, 0), radius 1)

• Angles: Measured in degrees (0° to 360°) or radians (0 to $2\pi$).

Example: The point (1, 0) lies on the unit circle because $1^2 + 0^2 = 1$.

Identifying Points on the Unit Circle

• To determine if a point $(x, y)$ is on the unit circle, substitute into $x^2 + y^2 = 1$.

• If the equation holds true, the point is on the circle.

Example: Is $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$ on the unit circle?

• Check: $\left( \frac{\sqrt{2}}{2} \right)^2 + \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{2}{4} + \frac{2}{4} = 1$

• Yes, it is on the unit circle.

Quadrants and Angle Location

• The coordinate plane is divided into four quadrants (Q1, Q2, Q3, Q4).

• Angles are measured from the positive x-axis, counterclockwise.

• To determine the quadrant of an angle in radians or degrees, compare its value to the boundaries of each quadrant.

Example: $\frac{7\pi}{4}$ radians is in Quadrant IV.

Sine, Cosine, and Tangent on the Unit Circle

Trigonometric Functions and the Unit Circle

Trigonometric functions relate angles to coordinates on the unit circle:

• Sine: $\sin \theta$ is the y-coordinate of the point on the unit circle at angle $\theta$.

• Cosine: $\cos \theta$ is the x-coordinate of the point on the unit circle at angle $\theta$.

• Tangent: $\tan \theta = \frac{\sin \theta}{\cos \theta}$

Example: For $\theta = 53^\circ$, the point on the unit circle is approximately $(0.6, 0.8)$, so $\cos 53^\circ \approx 0.6$, $\sin 53^\circ \approx 0.8$, $\tan 53^\circ \approx 1.33$.

Practice: Calculating Trigonometric Values

• Given a point on the unit circle, $(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})$, $\sin \theta = -\frac{\sqrt{2}}{2}$, $\cos \theta = -\frac{\sqrt{2}}{2}$, $\tan \theta = 1$.

Special Angles: 30°, 45°, and 60°

Memorizing Trig Values for Special Angles

The values of sine, cosine, and tangent for 30°, 45°, and 60° (or $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$ radians) are frequently used. There are two common methods to memorize them:

1. The 1-2-3 Rule: For $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$:

   • $\sin \theta = \frac{\sqrt{n}}{2}$, where n = 0, 1, 2, 3, 4 for $0^\circ$ to $90^\circ$.

   • $\cos \theta = \frac{\sqrt{n}}{2}$, but n counts in reverse order.

2. The Left Hand Rule: Use your left hand to count fingers for each angle; the number of fingers below or above the finger representing the angle gives the value for sine or cosine, respectively.

Table: Trig Values for Special Angles

Angle	$\sin \theta$	$\cos \theta$	$\tan \theta$
$30^\circ$ ($\frac{\pi}{6}$)	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^\circ$ ($\frac{\pi}{4}$)	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^\circ$ ($\frac{\pi}{3}$)	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

Reference Angles on the Unit Circle

Definition and Use of Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. Reference angles help relate any angle to a known angle in the first quadrant (Q1).

• To find the reference angle, measure the smallest angle between the terminal side and the x-axis.

• Reference angles are always positive and less than or equal to 90° ($\frac{\pi}{2}$ radians).

Example: The reference angle for 120° is 60°.

Trig Values in Quadrants II, III, and IV

Signs of Trigonometric Functions by Quadrant

The sign of sine, cosine, and tangent depends on the quadrant in which the angle's terminal side lies:

• Quadrant I (0° to 90°): All functions positive

• Quadrant II (90° to 180°): Sine positive, cosine and tangent negative

• Quadrant III (180° to 270°): Tangent positive, sine and cosine negative

• Quadrant IV (270° to 360°): Cosine positive, sine and tangent negative

Mnemonic: ASTC (All Students Take Calculus): All (I), Sine (II), Tangent (III), Cosine (IV) are positive in their respective quadrants.

Table: Signs of Trigonometric Functions by Quadrant

Quadrant	Sine	Cosine	Tangent
I	+	+	+
II	+	-	-
III	-	-	+
IV	-	+	-

Example: $\sin 150^\circ$ is positive (QII), $\cos 210^\circ$ is negative (QIII), $\tan 330^\circ$ is negative (QIV).

Summary

• The unit circle is a powerful tool for understanding trigonometric functions and their values for all angles.

• Special angles and their trig values should be memorized for quick reference.

• Reference angles and quadrant signs help extend knowledge of trig values to all angles.