The second page explores the graphical representation of sine and cosine functions when mapped from the unit circle to a coordinate system.

Definition: The sine function f$x$ = sin$x$ is created by plotting sine values against corresponding angles.

Highlight: Key properties include:

• Periodic with period 2π
• Value range between -1 and 1
• Zero points at multiples of π

Example: Special angle values are provided for both degree and radian measurements, showing the relationship between sine and cosine functions.

The third page details the general sine function and its transformations.

Definition: The general form of a sine function is f$x$ = a·sin$b(x-c)$ + d, where:

• a affects amplitude
• b affects period
• c affects horizontal shift
• d affects vertical shift

Example: Functions f₁$x$ = sin(2x) and f₂$x$ = sin(0.5x) demonstrate period changes.

Highlight: When a is negative, the graph is reflected across the x-axis.

The fourth page provides in-depth analysis of how parameters affect the sine function.

Definition: The period formula is given by b·p = 2π, where b is the parameter from the function equation.

Example: For f$x$ = -3sin$x+1$ + 0.5:

• Amplitude is 3
• Period is 2π
• Vertical shift is 0.5

Highlight: The parameter b doesn't directly indicate the stretch factor in the x-direction; this must be calculated separately.

The fifth page provides essential formulas and a comprehensive summary of sine function transformations.

Definition: The general sine function formula: f$x$ = a·sin$b(x-c)$ + d

Vocabulary: Key trigonometric ratios:

• sin(α) = opposite/hypotenuse
• cos(α) = adjacent/hypotenuse
• tan(α) = opposite/adjacent

Highlight: The period formula b·p = 2π can be used to find either the period p or parameter b.

The first page introduces fundamental concepts of trigonometric functions using the unit circle. The coordinates (u,v) of any point P on the unit circle correspond to cosine and sine values respectively.

Definition: The unit circle is a circle with radius 1 centered at the origin, where coordinates (u,v) represent (cos(α), sin(α)) for any angle α.

Example: For angle 60°, the coordinates are: u = cos(60°) = 0.5 v = sin(60°) = 0.866

Highlight: Trigonometric values repeat every 360°, leading to the relationship: sin$x + n·360°$ = sin$x$ and cos$x + n·360°$ = cos$x$

Vocabulary: The term "Einheitskreis" refers to the unit circle, which is fundamental for understanding trigonometric functions.