Circunferencia Trigonométrica - Ejercicios Resueltos - Nivel 1

Name: Circunferencia Trigonométrica - Ejercicios Resueltos - Nivel 1
Uploaded: 2016-01-04T13:28:55.000Z
Duration: 1 h 20 min 21 s

Introduction to Trigonometric Circles

Overview of Trigonometric Circles

Jorge introduces the topic of trigonometric circles, emphasizing their importance in solving problems related to angles and arcs.

A trigonometric circle is defined as a circle centered at the origin (0, 0), where the radius is always equal to one unit.

Key Points on Angles and Arcs

The point where the circle intersects with the positive x-axis is referred to as point A, known as the "origin of arcs."

To calculate the length of an arc, Jorge presents a formula: Length of Arc = Central Angle × Radius. For this circle, since the radius is 1, it simplifies to just the central angle.

Importance of Radians

Working with Radians

It’s crucial that all angles are expressed in radians when working with trigonometric functions; this ensures consistency in calculations.

Point M represents both the endpoint of an arc and the endpoint of an angle, which are significant terms often encountered in problems.

Quadrants and Their Characteristics

Understanding Quadrants

The Cartesian plane is divided into four quadrants. Each quadrant has specific characteristics regarding angles:

First Quadrant: From 0° to 90° (or 0 to π/2 radians).

Second Quadrant: From 90° to 180° (or π/2 to π radians).

Arc Length in Different Quadrants

In each quadrant, arcs correspond directly with their central angles:

First Quadrant: Arc length equals π/2.

Second Quadrant: Arc length from point A corresponds with its central angle up to π radians.

Third and Fourth Quadrants

Characteristics of Remaining Quadrants

The third quadrant spans from 180° (π radians) to 270° (3π/2 radians), where arc lengths also reflect these angles.

The fourth quadrant ranges from 270° (3π/2 radians) back around to a full rotation at 360° (or equivalently, back to zero).

Trigonometric Functions Across Quadrants

Positive Ratios by Quadrant

In the first quadrant, all trigonometric ratios (sin, cos, tan) are positive.

In the second quadrant, sine remains positive while cosine and tangent become negative; sine's reciprocal function is cosecant.

Trigonometric Functions in the Fourth Quadrant

Understanding Positive Trigonometric Ratios

In the fourth quadrant, the positive trigonometric ratios are cosine and its reciprocal, secant. The discussion begins with representing these functions on a unit circle.

Drawing Trigonometric Lines

A unit circle is drawn with its center at the origin. An angle alpha (α) is introduced, which has an arc extending from point A at the origin to a point with coordinates (0.8, 0.6).

Representing Sine of Angle Alpha

To represent sine(α), a perpendicular line is drawn from the endpoint of arc α down to the x-axis, illustrating that sine(α) corresponds to this vertical segment.

The length of this segment is determined to be 0.6, indicating that sine(α) = 0.6.

Exploring Angle Beta

The process for angle beta (β) starts similarly from point A and extends to another endpoint N.

Sine(β) is represented by drawing a perpendicular line from point N down to the x-axis.

Calculating Sine of Angle Beta

The coordinates for point N are (-0.6, -0.8). Thus, sine(β), represented by the y-coordinate, equals -0.8 due to being in the third quadrant where sine values are negative.

This reinforces that in the third quadrant only tangent and cotangent are positive; hence sine remains negative.

Cosine Representation

Drawing Cosine for Angle Alpha

To represent cosine(α), instead of drawing a line to the x-axis as done for sine, a perpendicular line is drawn towards the y-axis.

For angle α starting at point A and ending at M (coordinates: 0.8, 0.6), cosine(α) corresponds directly to its x-coordinate value of 0.8.

Cosine for Angle Beta

Similarly for angle β originating from point A and ending at N (coordinates: -0.6, -0.8), cosine(β)= -0.6 reflects its position in the third quadrant where cosine values are negative.

Tangent and Cotangent Representation

Tangent Line Construction

To illustrate tangent functions on a unit circle centered at origin with radius one, a tangent line is drawn through point Y that intersects perpendicularly with radius OA.

Tangent Function for Angle Alpha

For angle α extending from point A to another endpoint M on this tangent line illustrates how tangent can be visualized geometrically by prolonging side OA until it meets this tangent line.

This structured approach provides clarity on how trigonometric functions behave within different quadrants while emphasizing their geometric representations through angles alpha and beta along with their respective calculations in terms of sine, cosine, and tangent values based on their positions within those quadrants.

Understanding Tangents and Cotangents in Trigonometry

Tangent of Angle Alpha

The tangent of angle alpha is represented by segment AM. If the tangent line intersects above the x-axis, it has a positive value; if below, it has a negative value.

Tangent of Angle Beta

For angle beta, which terminates in the second quadrant, we extend the final side until it touches the tangent line at point N. Since point N is below the x-axis, the tangent for this angle will be negative.

Tangent of Angle Gamma

In contrast, for angle gamma located in the third quadrant, we again extend from point A to find where it meets the tangent line at point P. Here, since point P is above the x-axis, its tangent value is positive.

Tangent of Angle Theta

The process for angle theta involves extending its final side until it intersects with the tangent line in the fourth quadrant at point M. This segment also represents a positive value as per its position relative to the axes.

Representation of Cotangent Lines

To represent cotangent lines similar to tangents, we draw them from origin A instead of B (coordinates 0,y). We then follow similar steps as before to determine their values based on intersection points with respect to quadrants and axes positions.

Cotangent of Angle Alpha

The cotangent for angle alpha is determined by extending its arc until it meets a tangent line at point Y; if this intersection occurs rightward from y-axis, it's positive; leftward indicates negativity due to being in different quadrants.

Cotangent of Angle Beta

For angle beta's cotangent representation: after extending its arc into second quadrant and finding intersection with tangent line at some point Y (left side), this results in a negative cotangent value due to its position relative to y-axis.

Cotangent of Angle Gamma

In third quadrant for gamma's cotangent: upon extending and intersecting with tangent line at point P (right side), this yields a positive cotangent value as expected based on previous discussions about signs within respective quadrants.

Final Example - Cotangent of Angle Theta

The last example focuses on determining theta's cotangent through similar extension methods discussed earlier while ensuring clarity regarding sign conventions based on intersection points across various quadrants throughout these examples.

Understanding Secants and Co-secants in Trigonometry

Representation of Tangents and Secants

The tangent of angle theta is represented by extending the line from point B until it intersects with the tangent line at a specific point in the fourth quadrant.

The segment representing the tangent is denoted as segment D, which has a negative value since point Q lies to the left of the Y-axis.

Drawing Secant Lines

To represent secant lines, start by drawing a unit circle centered at the origin, marking points A and A' where they intersect with the X-axis.

A large diameter is drawn through point A', and then a secant for angle alpha is created by drawing a perpendicular line to this diameter.

The intersection of this perpendicular line with the diameter indicates that if it falls right of the Y-axis, it represents a positive secant; if left, it's negative.

Analyzing Different Arcs

For arc beta starting at the origin, its secant is represented similarly by drawing a perpendicular to its endpoint until it intersects with the X-axis. This results in segment N having a negative value as it lies on the left side of Y-axis.

Arc gamma's secant follows suit: draw a perpendicular from its endpoint to intersect X-axis at point P. Segment O-P will also be negative since it's located left of Y-axis.

Special Cases in Secants

When representing arc theta in quadrant four, again draw a perpendicular to find its secant. This leads us to segment O-Q.

Notably, certain angles like 90 degrees do not yield an intersection when attempting to draw their secants due to parallelism with axes—this occurs frequently with quadrantal angles (90°, 180°, etc.).

Introduction to Co-secant Lines

Transitioning from secants, co-secants are represented differently; instead of seeking intersections on X-axis, we look for them on Y-axis.

For arc alpha's co-secant representation, draw a perpendicular from its endpoint that intersects Y-axis at point Y. If above X-axis, it's positive; below yields negative values.

Conclusion on Co-secant Behavior

Understanding Trigonometric Functions in Different Quadrants

Representation of Secant and Cosecant

The secant of angle beta is represented by segment r, formed by drawing a perpendicular line from the terminal side to the y-axis.

For angle gamma, its cosecant is represented by segment o, which extends from the center of the canonical circle to the intersection point on the y-axis.

Angle theta in the fourth quadrant yields a negative cosecant value; a perpendicular line is drawn to intersect the y-axis at a specific point.

The process for determining secant and cosecant values remains consistent across different angles; viewers are encouraged to revisit earlier parts of the video if needed.

Problem Solving with Trigonometric Functions

Viewers are directed to download an exercise guide in PDF format that contains numerous problems for practice, starting with problem number one.

The first problem involves evaluating whether certain trigonometric statements (e.g., sine and cosine comparisons) are true or false using graphical representations on a unit circle.

Evaluating Sine Values

To compare sin(20°) and sin(70°), 20 degrees is marked on a protractor, establishing its corresponding arc on the unit circle.

A vertical line is drawn from point M (the endpoint of arc 20°) down to the x-axis to represent sin(20°).

Similarly, for sin(70°), another vertical line is drawn from point Y (the endpoint of arc 70°), allowing for direct comparison between both sine values.

Conclusion on Sine Comparisons

It’s concluded that sin(70°) is greater than sin(20°); thus, this statement about their relationship is deemed false as it contradicts initial assumptions.

Evaluating Cosine Values

The same method applies when comparing cosines: lines are drawn from endpoints of arcs downwards towards the y-axis for both angles.

Cosine values show that cos(20°)'s length exceeds that of cos(70°), confirming that cos(20°)>cos(70°).

Evaluating Tangent Values

To assess whether tan(20°)>tan(70°), tangent lines are extended from each angle's terminal side until they meet at a tangent line drawn parallel to the x-axis.

Understanding Tangents and Lengths in Geometry

Analysis of Tangent Lengths

The discussion begins with the comparison of tangent lengths, specifically focusing on the tangent of 70 degrees. It is illustrated through a geometric representation involving points A, N, and M.

The speaker asserts that the segment from point A to point N has a greater length than segment A to M, emphasizing that this is logically deduced based on their respective distances.

The conclusion drawn is that the tangent of 70 degrees represents a longer segment compared to others mentioned, leading to an assertion about its value being greater.

The statement regarding the tangential relationship is deemed false; thus, it leads to a clarification where true or false responses are discussed concerning previous assertions.