Precalculus Study Guide: Trigonometric Functions, Analytic Trigonometry, and Applications

Trigonometric Functions

Angle Measure: Radian and Degree

Angles can be measured in degrees or radians. One full rotation is 360° or $2\pi$ radians. The conversion between degrees and radians is given by:

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

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

Example: Convert 45° to radians: $45 \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.

The 6 Trigonometric Functions

The six fundamental trigonometric functions relate the angles of a right triangle to the ratios of its sides:

• Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$

• Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$

• Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)}$

Example: For $\theta = 30^\circ$, $\sin(30^\circ) = \frac{1}{2}$.

Reference Angles and Quadrants

The reference angle is the smallest angle between the terminal side of a given angle and the x-axis. The sign of trigonometric functions depends on the quadrant:

• Quadrant I: All functions positive

• Quadrant II: Sine positive

• Quadrant III: Tangent positive

• Quadrant IV: Cosine positive

Unit Circle and Angular Speed

The unit circle is a circle of radius 1 centered at the origin. The coordinates $(x, y)$ on the unit circle correspond to $(\cos(\theta), \sin(\theta))$ for an angle $\theta$.

• Angular speed: $\omega = \frac{\theta}{t}$, where $\theta$ is the angle in radians and $t$ is time.

Example: If a wheel rotates $200^\circ$ in 12 seconds, convert $200^\circ$ to radians: $200 \times \frac{\pi}{180} = \frac{10\pi}{9}$ radians. Angular speed: $\omega = \frac{10\pi/9}{12}$ radians/sec.

Analytic Trigonometry

Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable.

• Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$

• Quotient Identities: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

• Reciprocal Identities: $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, $\cot(\theta) = \frac{1}{\tan(\theta)}$

Inverse Trigonometric Functions

Inverse trigonometric functions allow you to find an angle given a trigonometric ratio.

• arcsin: $\arcsin(x)$ gives the angle whose sine is $x$

• arccos: $\arccos(x)$ gives the angle whose cosine is $x$

• arctan: $\arctan(x)$ gives the angle whose tangent is $x$

Example: $\arcsin\left(-\frac{1}{2}\right) = -30^\circ$

Solving Trigonometric Equations

To solve equations involving trigonometric functions, use identities and algebraic manipulation.

• Isolate the trigonometric function

• Apply inverse functions

• Consider all possible solutions within the given interval

Example: Solve $\sin(2\theta) = \frac{1}{2}$ for $90^\circ < 2\theta < 180^\circ$.

Applications of Trigonometric Functions

Ferris Wheel Problem

Trigonometric functions can model circular motion, such as a Ferris wheel. Key concepts include arc length, angular speed, and height above ground.

• Arc length: $s = r\theta$

• Angular speed: $\omega = \frac{\theta}{t}$

• Height above ground: Use the vertical component $y = r\sin(\theta)$

Example: For a 50 ft Ferris wheel rotated $200^\circ$, find the arc length: $s = 50 \times \frac{10\pi}{9}$

Systems of Equations and Matrices

Solving Systems of Equations

Systems of linear equations can be solved using substitution or elimination methods.

• Substitution: Solve one equation for a variable, substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

Example: Solve $x + y = 2$, $2x - y = 4$ by elimination.

Augmented Matrices

An augmented matrix represents a system of equations in matrix form, facilitating solution by row operations.

• Write coefficients and constants in matrix form

• Apply row operations to solve

Example: $\begin{bmatrix} 2 & 7 & | & 12 \\ 3 & 4 & | & 11 \end{bmatrix}$

Further Topics in Algebra

Partial Fraction Decomposition

Partial fraction decomposition expresses a rational function as a sum of simpler fractions, useful for integration and solving equations.

• Factor the denominator

• Set up an equation with unknown coefficients

• Solve for coefficients

Example: $\frac{2x+1}{x^2+x}$ can be decomposed as $\frac{A}{x} + \frac{B}{x+1}$

Summary Table: Trigonometric Function Signs by Quadrant

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

Practice Problems Overview

• Find all six trigonometric functions for given angles and points

• Convert between radians and degrees

• Use reference angles and determine quadrant signs

• Apply trigonometric identities and solve equations

• Model real-world scenarios with trigonometric functions

• Solve systems of equations using substitution and elimination

• Express systems as augmented matrices

• Perform partial fraction decomposition

Additional info: Some context and explanations have been expanded for clarity and completeness based on standard Precalculus curriculum.