Precalculus Study Notes: Trigonometric Functions, Identities, and Transformations

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 used to describe periodic phenomena and solve geometric problems.

• Sine ($\sin \theta$): Ratio of the length of the side opposite angle $\theta$ to the hypotenuse. $\sin \theta = \frac{O}{H}$

• Cosine ($\cos \theta$): Ratio of the length of the adjacent side to the hypotenuse. $\cos \theta = \frac{A}{H}$

• Tangent ($\tan \theta$): Ratio of the opposite side to the adjacent side. $\tan \theta = \frac{O}{A}$

• Cosecant ($\csc \theta$): Reciprocal of sine. $\csc \theta = \frac{H}{O}$

• Secant ($\sec \theta$): Reciprocal of cosine. $\sec \theta = \frac{H}{A}$

• Cotangent ($\cot \theta$): Reciprocal of tangent. $\cot \theta = \frac{A}{O}$

Example: In a right triangle with hypotenuse $H = 1$, the trigonometric ratios simplify to the lengths of the other sides.

Degrees and Radians

Conversion Between Units

Angles can be measured in degrees or radians. Radians are the standard unit in higher mathematics and are defined based on the arc length of a circle.

• Degrees: A full circle is 360 degrees.

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

• Conversion Formula: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

Example: $90^\circ = 90 \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

Trigonometric Identities

Fundamental Identities and Validity

Trigonometric identities are equations involving trigonometric functions that hold 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: This is NOT a valid identity; the correct reciprocal is $\tan \theta = \frac{1}{\cot \theta}$ or $\tan \theta = \frac{\sin \theta}{\cos \theta}$)

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

Example: To check validity, substitute values or use algebraic manipulation.

Transformations of Functions

Types and Effects on Graphs

Transformations alter the appearance of a function's graph without changing its fundamental nature. Common transformations include shifts, stretches, and reflections.

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
$cf(x)$	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

Example: $f(x) = (x-2)^2 + 3$ is a parabola shifted right by 2 units and up by 3 units.

Graphing Trigonometric Functions

General Form and Parameters

Trigonometric functions such as sine and cosine can be transformed to model various periodic phenomena. The general form for a sine function is:

• General Form: $f(x) = A \sin(B(x - a)) + C$

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

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

• Horizontal Shift ($a$): The amount the graph is shifted left or right.

• Vertical Shift ($C$): The amount the graph is shifted up or down.

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

Summary of Learning Outcomes

• Defined the basic trigonometric functions and their relationships.

• Converted between degrees and radians.

• Applied transformations to functions and understood their graphical effects.