LE COURS : Trigonométrie - Première

Introduction to Trigonometry

Overview of the Course

• The video aims to review key concepts in trigonometry, including trigonometric circles, radians, sine and cosine functions, and their applications for exams.

• Emphasis is placed on the necessity of practicing numerous exercises beyond this overview.

Introduction to the Trigonometric Circle

• A new geometric object called the "trigonometric circle" is introduced as a foundational element for understanding trigonometry.

• The concept of "direct sense" or "positive direction" is explained as counterclockwise rotation around the circle.

Understanding Directions and Definitions

Defining Direct Sense

• The direct sense refers to movement counterclockwise on the trigonometric circle, which is essential for defining angles.

• The trigonometric circle has its center at (0, 0) with a radius of 1 unit; this setup aids in establishing a coordinate system.

Associating Points on the Circle

• Each point on a tangent line can be uniquely associated with a point on the circle, facilitating circular measurements from linear references.

• This association allows for easier navigation between straight lines and circular paths in calculations.

Measuring Angles: Degrees vs Radians

Need for Radian Measurement

• Traditional degree measurement may not suffice; thus, radians are introduced as an angle measurement based on π (pi).

• One radian corresponds to an angle that intercepts an arc length equal to the radius of the circle (which is 1). A diagram illustrates this concept clearly.

Circumference and Angle Relationships

• The circumference of the unit circle is calculated as 2pi, indicating that one complete revolution equals 2pi radians or 360 degrees. Thus, there’s a direct relationship between degrees and radians established through proportionality.

Establishing Correspondences Between Degrees and Radians

Creating a Conversion Table

• A table correlates common angles in degrees with their radian equivalents:

• 360° = 2pi radians

• 180° = pi radians

• 90° = pi/2 radians

This systematic approach helps solidify understanding of angular measures within both systems.

Understanding Trigonometric Circles and Angle Measures

The Concept of Angle Measures on the Trigonometric Circle

• The challenge of reading angles on a trigonometric circle is highlighted, where multiple radian measures can correspond to a single position on the circle.

• An example illustrates that as one continues around the circle, they can reach positions like 11pi/4 , which corresponds back to earlier angles such as 3pi/4 .

• It is noted that rotating in both positive (counterclockwise) and negative (clockwise) directions leads to the same point, demonstrating the periodic nature of angle measures.

• The equivalence of angles is emphasized; for instance, 3pi/4 and -5pi/4 represent the same point on the circle.

• This results in an infinite number of measures for defining a single point on the trigonometric circle.

Establishing Principal Angle Measures

• To standardize discussions about angles, a unique principal measure is established for each angle, ensuring clarity in communication.

• The principal measure of an oriented angle is defined as being unique and falling within specific bounds: between -pi (exclusive) and pi .

• For example, starting with an angle like 7pi/4 , adding or subtracting multiples of 2pi yields equivalent angles but not necessarily their principal measure.

• Calculating these equivalents shows how different representations can still refer to the same angular position through addition or subtraction of full rotations.

• A general rule emerges: any angle expressed as t + k(2pi) , where k is an integer, represents equivalent angular measures.

Identifying Principal Angles

• Among all possible measures for an angle, only one will be considered its principal measure if it lies within specified limits.

• An example demonstrates that while values like 23pi/4 exceed acceptable bounds, values such as -pi/4 , fall within them and thus qualify as principal measures.

Reading Cosine and Sine from the Trigonometric Circle

• Transitioning to practical applications, cosine and sine values are derived from points on the trigonometric circle corresponding to specific angles.

• A diagrammatic representation aids in understanding how points relate to cosine (x-coordinate at point M on the unit circle).

• By drawing perpendicular lines from point M onto axes, we establish relationships between triangle sides formed by these intersections and their respective sine/cosine values.

• Utilizing right triangles allows us to apply known formulas for calculating sine and cosine based on adjacent side lengths relative to hypotenuse lengths.

Trigonometric Functions and Their Properties

Understanding Sine and Cosine on the Unit Circle

• The sine function is defined in relation to the unit circle, where sin(x) corresponds to the y-coordinate of point M on the circle, while cos(x) corresponds to the x-coordinate.

• Both sine and cosine functions yield values between -1 and 1. This is due to their geometric representation on the unit circle, where projections onto axes cannot exceed these bounds.

• The cosine function can be visualized as a projection onto the x-axis from point M on the unit circle, ensuring that it remains within [-1, 1].

• A key identity in trigonometry states that cos^2(x) + sin^2(x) = 1 . Here, cos^2(x) denotes (cos(x))^2 , which is a common notation for squaring trigonometric functions.

• The relationship between angles shows that cos(-x) = cos(x) , indicating cosine is an even function. In contrast, sine behaves differently: sin(-x) = -sin(x), showing it is an odd function.

Periodicity of Trigonometric Functions

• Adding multiples of 2pi does not change the value of sine or cosine; thus, both functions are periodic with a period of 2pi.

• For example, adding 4pi, or any multiple thereof, results in returning to the same position on the unit circle for both sine and cosine.

Notable Values of Trigonometric Functions

• Important values include specific angles like pi/3 , where properties such as cos(pi/3) = 1/2 , need to be memorized for quick reference.

• Understanding these values requires familiarity with reading from the unit circle—a skill emphasized as crucial for mastering trigonometry.

Advanced Properties and Formulas

• Various formulas exist within trigonometry; however, it's more important to know how to derive them rather than memorize them verbatim.

• For instance, using identities like those involving shifts by π (e.g., relating cos(π + x)) helps understand symmetry properties in trigonometric functions.

Symmetry in Trigonometric Functions

• When examining relationships such as between cos(π + x), one finds they exhibit central symmetry about certain points—specifically that they are opposites: if one is positive then its counterpart will be negative.

This structured approach provides clarity into fundamental concepts surrounding trigonometric functions while linking back directly to specific moments in time for further exploration.

Understanding Cosine and Sine Functions

Basic Properties of Cosine and Sine

• The cosine function is defined as a continuous property, which can be derived from sinusoidal functions. Understanding these relationships helps in grasping the underlying mathematics.

• Both cosine and sine depend on an angle, making the angle a variable that can be associated with a function. This leads to defining functions like cos(x) and sin(x).

Graphical Representation

• The graphical representation of cosine (cos(x)) and sine (sin(x)) shows they are similar but shifted by π/2 radians. This shift indicates their phase difference.

• Recognizing certain symmetries in these graphs is crucial for understanding their behavior over intervals.

Periodicity of Functions

• Observing the wave-like nature of both functions reveals their periodicity; they repeat indefinitely, allowing us to study them over finite intervals.

• A single interval of length π suffices to understand the behavior of these functions due to their periodic nature.

Function Characteristics

• The periodicity implies that every 2π units, the same values recur. Thus, studying within one period provides insights into all repetitions.

• Cosine and sine are classified as periodic functions with a period of 2π, meaning their patterns repeat every 2π radians.

Symmetry and Parity

• The properties of evenness (cosine being an even function: cos(-x)=cos(x)) and oddness (sine being an odd function: sin(-x)=-sin(x)) help in analyzing their graphs.

• Even functions exhibit symmetry about the y-axis while odd functions show symmetry about the origin.

Interval Study Simplification

• By focusing on specific intervals such as [0, π], we can simplify our analysis without losing generality due to symmetry properties.

• This approach allows for efficient graphing since once one side is plotted, it can be mirrored across axes based on parity.

Conclusion on Function Behavior

• Analyzing only positive intervals simplifies studies further; negative values can be inferred through symmetry.

• Ultimately, studying cosine or sine over [0, π] captures essential characteristics needed for comprehensive understanding while leveraging symmetry for full graph representation.