Reduction Formulas and Function Signs

This page focuses on wzory redukcyjne (reduction formulas) and the signs of trigonometric functions in different quadrants. Reduction formulas are essential tools for simplifying trigonometric expressions and solving equations.

The page presents a comprehensive table of reduction formulas for sine, cosine, tangent, and cotangent functions. These formulas show how to express trigonometric functions of angles greater than 90° in terms of functions of acute angles.

Vocabulary: Reduction formulas are trigonometric identities that relate the trigonometric functions of an angle to those of a smaller angle.

A detailed chart illustrates the signs of trigonometric functions in each quadrant, reinforcing the information from the previous page. This visual aid is particularly useful for quickly determining the sign of a function for any given angle.

Highlight: The reduction formulas and sign chart are crucial for solving problems involving funkcje trygonometryczne kąta rozwartego (trigonometric functions of obtuse angles).

Periodicity and Parity of Trigonometric Functions

This page delves into two important properties of trigonometric functions: periodicity and parity. These properties are fundamental in understanding the behavior of trigonometric functions and are crucial for solving more complex trigonometric equations.

The concept of periodicity is introduced, defining a periodic function and providing the period for each of the main trigonometric functions.

Definition: A function f$x$ is periodic with period T if f$x + T$ = f$x$ for all x in the domain of f.

The periods for sine, cosine, tangent, and cotangent functions are given:

• Sine and cosine have a period of 2π
• Tangent and cotangent have a period of π

The parity of trigonometric functions is also explained, distinguishing between even and odd functions.

Vocabulary: An even function is symmetric about the y-axis, while an odd function is symmetric about the origin.

The page provides the parity for each trigonometric function:

• Cosine is an even function: cos$-x$ = cos$x$
• Sine, tangent, and cotangent are odd functions: sin$-x$ = -sin$x$, tan$-x$ = -tan$x$, cot$-x$ = -cot$x$

These properties are essential for understanding tożsamości trygonometryczne (trigonometric identities) and solving more advanced trigonometric problems.

Angle Measurement and Trigonometric Values

This final page covers two crucial aspects of trigonometry: angle measurement and common trigonometric values. It introduces the concept of radian measure and provides conversion formulas between degrees and radians.

Definition: The radian is the standard unit of angular measure in many areas of mathematics. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

The page presents formulas for converting between degrees and radians:

• Degrees to radians: α [rad] = (α° · π) / 180°
• Radians to degrees: α° = (α [rad] · 180°) / π

Highlight: Understanding miara łukowa kąta (radian measure) is crucial for advanced mathematics and physics applications.

The page concludes with a table of wartości trygonometryczne (trigonometric values) for common angles (0°, 30°, 45°, 60°, 90°). This tabela wartości funkcji trygonometrycznych (table of trigonometric function values) is an essential reference for solving trigonometric problems without a calculator.

Example: At 45°, sin(45°) = cos(45°) = √2/2, and tan(45°) = 1

This information is crucial for obliczanie wartości funkcji trygonometrycznych (calculating trigonometric function values) and solving various trigonometric problems.

Trigonometric Functions and Quadrants

This page introduces the fundamental funkcje trygonometryczne (trigonometric functions) and their behavior in different quadrants of the coordinate system. The sine, cosine, tangent, and cotangent functions are defined in relation to the unit circle.

Definition: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system.

The page also presents the quadrant divisions and the signs of trigonometric functions in each quadrant.

Highlight: In the first quadrant, all trigonometric functions are positive. In the second quadrant, only sine is positive. In the third quadrant, tangent and cotangent are positive. In the fourth quadrant, only cosine is positive.

Important trigonometric identities are also provided, including the Pythagorean identity (sin²α + cos²α = 1) and relationships between tangent, cotangent, sine, and cosine.

Example: The tangent function can be expressed as the ratio of sine to cosine: tan α = sin α / cos α