Graphs of Trigonometric Functions

This page focuses on the graphical representations of trigonometric functions, particularly the tangent and cotangent functions. It provides detailed information about their properties and behaviors.

The tangent function, f(α) = tan(α), is discussed in depth:

• It has a period of π or 180°
• The function is undefined at 90° and 270°
• It exhibits point symmetry about the origin

Similarly, the cotangent function, f(α) = cot(α), is described:

• It also has a period of π
• It is undefined at multiples of 180°
• It also shows point symmetry about the origin

Vocabulary: Definitionslücke refers to the points where the function is undefined, which occur at regular intervals for both tangent and cotangent functions.

Highlight: The graphs of these functions are provided, clearly showing their periodic nature and the points where they approach infinity.

The page also includes important trigonometric identities related to these functions, such as tan(180° - α) = -tan(α) and cot(180° + α) = cot(α).

Transformations of Trigonometric Functions

This page explores how the basic sine and cosine functions can be transformed to create more complex trigonometric functions. It focuses on the general form f(x) = a · sin(b · x + c) + d, explaining the effect of each parameter.

The transformations covered include:

• Vertical shifts (controlled by parameter d)
• Horizontal shifts (controlled by parameter c)
• Vertical stretching or compression (controlled by parameter a)
• Horizontal stretching or compression (controlled by parameter b)

Example: The page provides examples of each transformation, such as f(x) = sin(x) - 1 for a vertical shift downward by 1 unit.

Highlight: Understanding these transformations is crucial for modeling periodic phenomena in various scientific and engineering applications.

The page also discusses how negative values of parameters can lead to reflections of the function graph:

• A negative value of a results in a reflection about the x-axis
• A negative value of b results in a reflection about the y-axis

Vocabulary: Streckung refers to stretching, while Stauchung refers to compression of the function graph.

Pythagorean Theorem and Its Applications

This page focuses on the Satz des Pythagoras anwenden (application of the Pythagorean theorem) and its various forms for solving problems involving right-angled triangles.

The Pythagorean theorem is presented in its standard form:

c² = a² + b²

Where c is the hypotenuse, and a and b are the other two sides of a right-angled triangle.

Highlight: The page emphasizes that the Pythagorean theorem only applies to right-angled triangles.

The page also provides rearranged forms of the theorem for finding the length of either cathetus:

a² = c² - b² b² = c² - a²

Example: The page demonstrates how to use the Pythagorean theorem to calculate unknown side lengths in a right-angled triangle.

Additionally, the page covers practical applications of trigonometry, such as calculating slopes and angles:

• Using tangent to find the angle of inclination for a given slope
• Using inverse sine (arcsin) to find an angle when given the opposite side and hypotenuse

Vocabulary: Steigungswinkel refers to the angle of inclination or slope angle.

The page concludes with a step-by-step approach for solving problems using the Pythagorean theorem:

1. Draw a sketch
2. Note and label given information
3. Calculate unknown quantities
4. Write a complete answer sentence