Trigonometric Functions: Sine & Cosine

Parent Functions and Key Properties

The sine and cosine functions are fundamental periodic functions in trigonometry, used to model cycles and waves. Their parent functions have specific properties:

• Amplitude: 1

• Period:

• Midline:

• Domain:

• Range:

Key Points:

• Sine starts at the midline ().

• Cosine starts at a maximum ().

• One full cycle occurs every units.

Transformations of Sine & Cosine

Transformations allow us to modify the basic sine and cosine graphs. The general forms are:

Parameter Meanings:

• A: Amplitude ()

• B: Period ()

• C: Phase shift ()

• D: Vertical shift (midline )

Transformation Effects:

• Vertical stretch/shrink:

• Reflection over x-axis:

• Horizontal stretch/shrink:

• Phase shift:

• Vertical shift:

Graphing Sine & Cosine Functions

To graph transformed sine or cosine functions, follow these steps:

1. Find the midline ().

2. Identify the amplitude ().

3. Calculate the period ().

4. Divide the period into four equal parts.

5. Plot key points and sketch the curve.

Notes:

• Sine crosses the midline at the start.

• Cosine starts at a maximum or minimum.

• The graph repeats every period.

Writing an Equation from a Graph

To write the equation for a sine or cosine graph:

1. Find the amplitude: half the distance between maximum and minimum values.

2. Find the midline: average of maximum and minimum values.

3. Determine the period: horizontal distance of one full cycle.

4. Decide whether the graph is sine or cosine based on starting point and shape.

5. Write the equation using the general form.

Tip: Always find amplitude and midline first.

Tangent, Cotangent, Secant, and Cosecant Functions

Tangent Function

The tangent function is another fundamental trigonometric function:

• Parent Function:

• Period:

• Range:

• Domain: Excludes vertical asymptotes at

General Form:

Cotangent Function

• Parent Function:

• Period:

• Vertical Asymptotes:

• Graph: Decreasing curve

Secant & Cosecant Functions

• Secant:

• Cosecant:

• Vertical Asymptotes: Where or

• Range: or

• Graph: U-shaped repeating curves

Solving Trigonometric Equations Using Graphs

Graphical Solution Steps

Graphing can be used to solve trigonometric equations, especially when algebraic methods are difficult:

1. Graph both sides of the equation.

2. Identify intersection points.

3. Solutions are the x-values of intersections.

When to Use:

• When equations are hard to solve algebraically.

• When approximate answers are acceptable.

Inverse Trigonometric Functions

Purpose and Properties

Inverse trigonometric functions "undo" the original trigonometric functions. Their domains are restricted to ensure they are functions.

Evaluating Inverse Trig Expressions

• Exact Values: Use the unit circle or special right triangles. Answers must be in the inverse function’s range.

• Example:

• Composite Expressions: ; only works if is in the principal range.

Applications: Right Triangles & Bearings

Solving a Right Triangle

To solve a right triangle, find all missing sides and angles using trigonometric ratios and the Pythagorean Theorem:

• Use SOH CAH TOA for sine, cosine, and tangent ratios.

• Use the Pythagorean Theorem:

Bearings

• Bearings are measured from North or South.

• Always draw a diagram to visualize the problem.

• Use trigonometric ratios to solve for unknowns.

Simple Harmonic Motion & Multi-Triangle Problems

Simple Harmonic Motion

Simple harmonic motion is modeled by a sine function:

• Model:

• A: Amplitude

• B: Angular speed

• D: Vertical shift

Applications: Ferris wheels, springs, tides

Multiple Triangle Problems

• Break the figure into triangles.

• Solve one triangle at a time.

• Use the Law of Sines or Law of Cosines when needed.

Law of Sines:

Law of Cosines: