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$α$.

Trigonometric Functions and the Unit Circle

This page delves into the relationship between trigonometric functions and the unit circle, introducing the concept of radian measure and its conversion from degrees.

Key concepts covered include:

• The definition of radian measure in terms of the circumference of a circle
• Conversion formulas between degrees and radians
• The unit circle and its relationship to sine and cosine functions

Definition: One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Example: The page provides examples of converting between degrees and radians, such as 35° = 0.61 radians.

The page also introduces the graphs of sine and cosine functions, highlighting their periodic nature and key properties:

• The amplitude and period of these functions
• The relationship between sine and cosine: cos$α$ = sin$α + 90°$
• Symmetry properties of sine and cosine functions

Highlight: The unit circle is a powerful tool for understanding trigonometric functions, as it visually represents the relationship between angles and their sine and cosine values.

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

Addition Theorems and Trigonometric Identities

This final page presents a comprehensive list of addition theorems and trigonometric identities, which are crucial for solving complex trigonometric equations and simplifying expressions.

Some of the key formulas presented include:

1. sin$α ± β$ = sin$α$ cos$β$ ± cos$α$ sin$β$
2. cos$α ± β$ = cos$α$ cos$β$ ∓ sin$α$ sin$β$
3. tan$α ± β$ = $tan(α$ ± tan$β$) / $1 ∓ tan(α$ tan$β$)

Highlight: These addition theorems are fundamental in trigonometry and have wide-ranging applications in physics and engineering.

The page also includes double angle formulas:

4. sin$2x$ = 2 sin$x$ cos$x$
5. cos$2x$ = cos²$x$ - sin²$x$ = 1 - 2sin²$x$ = 2cos²$x$ - 1

Example: The formula for tan$2x$ is given as tan$2x$ = $2 tan(x$) / $1 - tan²(x$).

Additional useful identities are provided, such as:

• Product-to-sum formulas for sine and cosine
• Sum-to-product formulas for sine and cosine

Vocabulary: Additionstheoreme refers to addition theorems in trigonometry.

The page concludes with a visual representation of how sine and cosine functions relate to each other through phase shifts, which is crucial for understanding their behavior and relationships.

Trigonometric Ratios and the Pythagorean Theorem

This page introduces the fundamental trigonometric ratios and their relationship to the sides of a right-angled triangle. It also demonstrates how to apply these concepts to solve practical problems.

The trigonometric ratios sine, cosine, and tangent are defined as follows:

• Sine $sin$: Opposite side / Hypotenuse
• Cosine $cos$: Adjacent side / Hypotenuse
• Tangent $tan$: Opposite side / Adjacent side

Definition: The Satz des Pythagoras $Pythagorean theorem$ states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

Example: A practical application is provided, involving a triangle with sides 4.4m and 5.8m. The problem demonstrates how to calculate angles using inverse trigonometric functions and how to find the length of the third side using the Pythagorean theorem.

Highlight: The page emphasizes the importance of understanding these ratios for solving problems involving right-angled triangles, which is crucial for many applications in physics and engineering.