Chapter 5: Trigonometric Functions – Review and Applications

Trigonometric Functions and Their Applications

5.1 Angles and Their Measurement

Understanding angles and their measurement is foundational in trigonometry. Angles can be measured in degrees or radians, and their properties are essential for solving geometric and trigonometric problems.

• Drawing Angles: To draw an angle, start from the initial side (usually the positive x-axis) and rotate counterclockwise for positive angles or clockwise for negative angles. The amount of rotation determines the angle's measure.

• Coterminal Angles: Coterminal angles share the same initial and terminal sides. To find coterminal angles, add or subtract multiples of $360^\circ$ (degrees) or $2\pi$ (radians):

  • In degrees: $\theta_{cot} = \theta + 360^\circ \times k$

  • In radians: $\theta_{cot} = \theta + 2\pi k$

  • where $k$ is any integer.

• Degree-Radian Conversion:

  • Degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

  • Radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

• Arc Length: The length $s$ of an arc of a circle with radius $r$ and central angle $\theta$ (in radians) is: $s = r\theta$

• Area of a Sector: The area $A$ of a sector with radius $r$ and central angle $\theta$ (in radians) is: $A = \frac{1}{2} r^2 \theta$

• Finding Unknowns: Given any two of arc length, radius, or central angle, use the above formulas to solve for the third.

• Example: Find the length of an arc with radius $5$and central angle $60^\circ$.

  • Convert $60^\circ$ to radians: $60^\circ \times \frac{\pi}{180} = \frac{\pi}{3}$

  • Arc length: $s = 5 \times \frac{\pi}{3} = \frac{5\pi}{3}$

5.2 Trigonometric Functions and the Unit Circle

The unit circle is a circle of radius 1 centered at the origin. It is a powerful tool for evaluating trigonometric functions and understanding their properties.

• Trigonometric Functions Using a Point: For a point $(x, y)$ on the unit circle corresponding to an angle $\theta$:

  • $\sin \theta = y$

  • $\cos \theta = x$

  • $\tan \theta = \frac{y}{x}$ (if $x \neq 0$)

  • $\csc \theta = \frac{1}{y}$ (if $y \neq 0$)

  • $\sec \theta = \frac{1}{x}$ (if $x \neq 0$)

  • $\cot \theta = \frac{x}{y}$ (if $y \neq 0$)

• Circle of Radius $r$: For a point $(x, y)$ on a circle of radius $r$:

  • $\sin \theta = \frac{y}{r}$

  • $\cos \theta = \frac{x}{r}$

  • $\tan \theta = \frac{y}{x}$

  • Other functions are reciprocals as above.

• Example: If a point $(3, 4)$ lies on a circle of radius $5$, then$\sin \theta = \frac{4}{5}$,$\cos \theta = \frac{3}{5}$, $\tan \theta = \frac{4}{3}$.

5.3 Evaluating Trigonometric Functions Given Partial Information

Given one trigonometric value and the quadrant of the angle, you can determine the remaining trigonometric functions.

• Process:

  1. Use the given value to find $x$, $y$, or $r$ as needed.

  2. Apply the Pythagorean identity: $x^2 + y^2 = r^2$.

  3. Determine the signs of $x$ and $y$ based on the quadrant.

  4. Calculate the remaining trigonometric functions.

• Example: If $\sin \theta = \frac{3}{5}$ and $\theta$ is in Quadrant II:

  • $y = 3$, $r = 5$

  • $x = -\sqrt{r^2 - y^2} = -\sqrt{25 - 9} = -4$ (negative in Quadrant II)

  • $\cos \theta = \frac{x}{r} = -\frac{4}{5}$

  • $\tan \theta = \frac{y}{x} = -\frac{3}{4}$

5.4 – 5.6 Graphs and Properties of Trigonometric Functions

Sinusoidal functions and their transformations are central to modeling periodic phenomena. Understanding their properties and graphs is essential in trigonometry.

• Properties of Sinusoidal Functions:

  • Amplitude: The maximum value from the midline. For $y = a \sin(bx + c) + d$, amplitude is $|a|$.

  • Period: The length of one cycle. For $y = a \sin(bx)$, period is $\frac{2\pi}{|b|}$.

  • Phase Shift: Horizontal shift, given by $-\frac{c}{b}$ in $y = a \sin(bx + c)$.

  • Vertical Shift: The value $d$ in $y = a \sin(bx + c) + d$.

• Graphing Sinusoidal Functions:

  • Identify amplitude, period, phase shift, and vertical shift.

  • Plot key points (maximum, minimum, intercepts) and sketch the curve.

  • Apply transformations in the correct order: horizontal shift, stretch/compression, reflection, vertical shift.

• Finding Equations from Graphs: Analyze the graph to determine amplitude, period, phase shift, and vertical shift, then write the equation in the form $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$.

• Graphing Tangent and Cotangent Functions:

  • Tangent: $y = a \tan(bx + c) + d$; period is $\frac{\pi}{|b|}$.

  • Cotangent: $y = a \cot(bx + c) + d$; period is $\frac{\pi}{|b|}$.

  • Vertical asymptotes occur where the function is undefined.

• Graphing Secant and Cosecant Functions:

  • Secant and cosecant are reciprocals of cosine and sine, respectively.

  • Graph the corresponding sine or cosine function first, then draw the reciprocal curves, placing vertical asymptotes where the original function is zero.

• Graphing with Phase Shift: For $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$, the graph is shifted horizontally by $-\frac{c}{b}$ units.

• Example: Graph $y = 2 \sin\left(3x - \frac{\pi}{2}\right) + 1$:

  • Amplitude: $2$

  • Period: $\frac{2\pi}{3}$

  • Phase shift: $\frac{\pi}{6}$ to the right

  • Vertical shift: $1$ up

Summary Table: Properties of Basic Trigonometric Functions

Function	Amplitude	Period	Phase Shift	Vertical Shift
$y = a \sin(bx + c) + d$	$|a|$	$\frac{2\pi}{|b|}$	$-\frac{c}{b}$	$d$
$y = a \cos(bx + c) + d$	$|a|$	$\frac{2\pi}{|b|}$	$-\frac{c}{b}$	$d$
$y = a \tan(bx + c) + d$	n/a	$\frac{\pi}{|b|}$	$-\frac{c}{b}$	$d$

Additional info: This guide expands on the review points by providing definitions, formulas, and examples for each topic, ensuring a comprehensive understanding suitable for Precalculus students.