Trigonometric Functions and Transformations: Precalculus Study Notes

Trigonometric Functions

Definitions and Relationships

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. These functions are fundamental in precalculus and are widely used in mathematics, physics, and engineering.

• Sine (sin): $\sin \theta = \frac{O}{H}$, where O is the length of the side opposite angle $\theta$, and H is the hypotenuse.

• Cosine (cos): $\cos \theta = \frac{A}{H}$, where A is the length of the adjacent side.

• Tangent (tan): $\tan \theta = \frac{O}{A}$

• Cosecant (csc): $\csc \theta = \frac{H}{O}$

• Secant (sec): $\sec \theta = \frac{H}{A}$

• Cotangent (cot): $\cot \theta = \frac{A}{O}$

Note: If the hypotenuse has length 1 (unit circle), these ratios simplify to the coordinates on the unit circle.

Trigonometric Identities

Fundamental Identities

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

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

• Co-function Identity: $\sin \theta = \cos \left( \frac{\pi}{2} - \theta \right)$

• Reciprocal Identity: $\tan \theta = \frac{1}{\csc \theta}$ (Note: The correct reciprocal identity is $\tan \theta = \frac{1}{\cot \theta}$ or $\tan \theta = \frac{\sin \theta}{\cos \theta}$)

• Double Angle Identity for Cosine: $\cos^2 \theta = \frac{1}{2}(1 + \cos(2\theta))$

Example: Which of the following is not a valid trigonometric identity?

• 1) $\sin^2 \theta + \cos^2 \theta = 1$

• 2) $\sin \theta = \cos \left( \frac{\pi}{2} - \theta \right)$

• 3) $\tan \theta = \frac{1}{\csc \theta}$ (Not valid)

• 4) $\cos^2 \theta = \frac{1}{2}(1 + \cos(2\theta))$

Angle Measurement

Degrees and Radians

Angles can be measured in degrees or radians. Radians are the standard unit in higher mathematics.

• Degrees: A full circle is 360°.

• Radians: A full circle is $2\pi$ radians.

• Conversion: $1\text{ radian} = \frac{180}{\pi}$ degrees

Example: To convert 90° to radians: $90° \times \frac{\pi}{180°} = \frac{\pi}{2}$ radians.

Transformations of Functions

Types of Transformations

Transformations change the position or shape of a function's graph. Understanding these is essential for graphing and analyzing functions.

Transformation of $f(x)$	Effect on the graph of $f$
$f(x) + c$	Vertical shift up $c$ units
$f(x) - c$	Vertical shift down $c$ units
$f(x + c)$	Shift left by $c$ units
$f(x - c)$	Shift right by $c$ units
$c f(x)$	Vertical stretch if $c > 1$; vertical compression if $0 < c < 1$
$f(cx)$	Horizontal stretch if $0 < c < 1$; horizontal compression if $c > 1$
$-f(x)$	Reflection about the $x$-axis
$f(-x)$	Reflection about the $y$-axis

Graphing Trigonometric Functions

General Form and Parameters

The general form for a sine or cosine function is:

$f(x) = A \sin(B(x - a)) + C$

• Amplitude (A): The height from the center line to the peak.

• Period: The length of one complete cycle, given by $\frac{2\pi}{|B|}$.

• Horizontal Shift (a): The graph shifts right by $a$ units if $a > 0$.

• Vertical Shift (C): The graph shifts up by $C$ units.

Example: For $f(x) = 3\sin(2(x - \frac{\pi}{4})) + 1$, the amplitude is 3, period is $\pi$, horizontal shift is $\frac{\pi}{4}$ to the right, and vertical shift is 1 up.

Summary of Key Learning Outcomes

• Define the basic trigonometric functions and their relationships.

• Convert between degrees and radians for angle measurement.

• Apply and interpret transformations to functions, including trigonometric functions.