Unit Circle and Graphs of Trigonometric Functions

Unit Circle and Trigonometric Functions

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin of the coordinate plane. It is used to define the trigonometric functions for all real numbers and to visualize their values.

• Key Points: The unit circle allows us to relate angles (measured in radians or degrees) to coordinates on the circle.

• Coordinates: Any point on the unit circle corresponding to an angle θ has coordinates $(\cos \theta, \sin \theta)$.

• Quadrants: The signs of sine and cosine values depend on the quadrant in which the terminal side of the angle lies.

• Reference Angles: Reference angles help determine the values of trigonometric functions for angles outside the first quadrant.

Example: The point on the unit circle at $\theta = 45^\circ$ (or $\frac{\pi}{4}$ radians) is $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Graphing Sine and Cosine Functions

The graphs of the sine and cosine functions are periodic and oscillate between -1 and 1. These graphs are essential for understanding wave behavior and periodic phenomena.

• Sine Function: $y = \sin x$

• Cosine Function: $y = \cos x$

• Period: Both functions have a period of $2\pi$.

• Amplitude: The maximum value is 1, and the minimum value is -1.

• Key Points: For $y = \sin x$, the function passes through the origin (0,0). For $y = \cos x$, the function starts at (0,1).

Example: To graph $y = \sin x$, plot points at $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$ with corresponding $y$-values $0, 1, 0, -1, 0$.

Table of Sine and Cosine Values

Tables are often used to summarize the values of sine and cosine for commonly used angles. This helps in quickly referencing values when graphing or solving equations.

Angle (Degrees)	Angle (Radians)	$\sin \theta$	$\cos \theta$
0°	0	0	1
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$
90°	$\frac{\pi}{2}$	1	0

Additional info: The table above is inferred from standard trigonometric values for common angles, as the image contains a partially visible table of values.

Plotting and Connecting Points

When graphing trigonometric functions, plot the calculated points for each angle and connect them with a smooth, continuous curve to reveal the periodic nature of the sine and cosine functions.

• Plot points for several key angles (as shown in the table above).

• Draw a smooth curve through the points to complete one period of the function.

Example: For $y = \sin x$, plot points at $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$ and connect them smoothly.