Chapter 9.5: Circular Functions – Definitions, Evaluation, and Applications

Trigonometry Functions and Applications

Introduction to Circular Functions

Circular functions are the extension of trigonometric functions to all real numbers, using the unit circle. These functions are fundamental in precalculus and are widely used in modeling periodic phenomena.

• Trigonometric functions (sine, cosine, tangent, etc.) become circular functions when defined for all real values.

• The unit circle is a circle centered at the origin (0,0) with radius 1, used to define these functions for any real number.

• Arc length s is measured from the point (1,0) counterclockwise for s > 0 and clockwise for s < 0.

Definitions of Circular Functions

For any real number s, let (x, y) be the endpoint of the arc of length s on the unit circle $x^2 + y^2 = 1$. The circular functions are defined as:

• Sine: $\sin s = y$

• Cosine: $\cos s = x$

• Tangent: $\tan s = \frac{y}{x}$   (where $x \neq 0$)

• Cosecant: $\csc s = \frac{1}{y}$   (where $y \neq 0$)

• Secant: $\sec s = \frac{1}{x}$   (where $x \neq 0$)

• Cotangent: $\cot s = \frac{x}{y}$   (where $y \neq 0$)

Evaluating Circular Functions

Circular function values for real numbers are found similarly to trigonometric function values for angles in radians. Calculators must be set to radian mode when evaluating these functions.

• Exact values can be found using reference angles and properties of special triangles.

• Decimal approximations are obtained using calculators.

Examples: Evaluating Circular Functions

Example 1: $s = \frac{3\pi}{2}$

• $\sin \frac{3\pi}{2} = -1$

• $\cos \frac{3\pi}{2} = 0$

• $\tan \frac{3\pi}{2}$ is undefined

Example 2: $s = \frac{7\pi}{4}$

• Point on unit circle: $x = \frac{\sqrt{2}}{2}$, $y = -\frac{\sqrt{2}}{2}$

• $\sin \frac{7\pi}{4} = -\frac{\sqrt{2}}{2}$

• $\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}$

• $\tan \frac{7\pi}{4} = -1$

Example 3: $s = \frac{5\pi}{6}$

• Point on unit circle: $x = -\frac{\sqrt{3}}{2}$, $y = \frac{1}{2}$

• $\sin \frac{5\pi}{6} = \frac{1}{2}$

• $\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$

• $\tan \frac{5\pi}{6} = -\frac{\sqrt{3}}{3}$

• $\csc \frac{5\pi}{6} = 2$

• $\sec \frac{5\pi}{6} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}$

• $\cot \frac{5\pi}{6} = -\sqrt{3}$

Special Angles and the Unit Circle

Special angles (multiples of $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$) have well-known coordinates on the unit circle, which are used to quickly evaluate circular functions.

Finding Circular Function Values

• Use the unit circle to find exact values for sine, cosine, and tangent at special angles.

• For example, $\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}$, $\sin \frac{7\pi}{4} = -\frac{\sqrt{2}}{2}$.

• Angles coterminal with special angles can be evaluated using their reference angle.

• Calculator approximations: $\cos(1.85) \approx -0.2756$ (in radian mode).

Inverse Circular Functions

To find the arc length s given a circular function value, use the inverse function. For example, if $\cos s = 0.457$, then $s = \cos^{-1}(0.457) \approx 1.096$ (radians).

Applications of Circular Functions

Modeling the Moon's Phase

• The phase F of the moon is modeled by $F(t) = \frac{1}{2}(1 - \cos t)$, where t is the phase angle.

• This function gives the fraction of the moon's face illuminated by the sun.

Phase Angle (t)	Moon Phase	Illumination Fraction
$0$	New Moon	$0$
$\frac{\pi}{2}$	First Quarter	$\frac{1}{2}$
$\pi$	Full Moon	$1$
$\frac{3\pi}{2}$	Last Quarter	$\frac{1}{2}$

Modeling Tides

• Tide levels can be modeled by a cosine function: $f(x) = 0.6 \cos[0.511(x - 2.4)] + 2$

• Time between high tides is the period between peaks (e.g., 12.3 hours).

• Difference between high and low tide is the amplitude (e.g., 1.2 feet).

• Example: $f(10) = 0.6 \cos[0.511(10 - 2.4)] + 2 \approx 1.56$ feet

Right Triangle Ratios and Circular Functions

Trigonometric ratios in right triangles correspond to circular functions on the unit circle:

Function	Triangle Ratio
$\cos \theta$	$\frac{\text{adjacent}}{\text{hypotenuse}}$
$\sin \theta$	$\frac{\text{opposite}}{\text{hypotenuse}}$
$\tan \theta$	$\frac{\text{opposite}}{\text{adjacent}}$
$\sec \theta$	$\frac{\text{hypotenuse}}{\text{adjacent}}$
$\csc \theta$	$\frac{\text{hypotenuse}}{\text{opposite}}$
$\cot \theta$	$\frac{\text{adjacent}}{\text{opposite}}$

Example: Finding Lengths of Line Segments

Given a right triangle with angles $30^\circ$ and $60^\circ$, the lengths of segments can be found using circular function values:

• $OQ = \cos 30^\circ = \frac{\sqrt{3}}{2}$

• $PQ = \sin 30^\circ = \frac{1}{2}$

• $VR = \tan 30^\circ = \frac{\sqrt{3}}{3}$

• $OV = \sec 30^\circ = \frac{2\sqrt{3}}{3}$

• $OU = \csc 30^\circ = 2$

• $US = \cot 30^\circ = \sqrt{3}$

Summary: Circular functions generalize trigonometric functions to all real numbers using the unit circle, allowing for the modeling and analysis of periodic phenomena in mathematics and science.