TRIGONOMETRY

SEMEPXDOST 2020: Mathematics Presentation

Introduction to Presentation

• Presentation conducted by UPLB DOST Scholars' Society.

How to Use This Video

• Purpose of the video includes:
  • Refreshing your memory of content.
  • Familiarizing you with how the topic might be incorporated into this year's entrance tests and DOST exam.
  • Supporting your own in-depth review of Trigonometry (not a replacement).

Things to Keep in Mind

• Acknowledgment that mistakes may happen and should not be discouraged.
• Encouragement to ask questions or raise clarifications.
• Emphasis on the importance of mental health over exam preparation.

Topics Covered

1. Basics of Trigonometry
2. Graphs of Trigonometric Functions
3. Angles of Elevation and Depression
4. Basic Trigonometric Identities
5. Triangle Complexities
6. Trigonometric Equations and Inverses
7. Solved Problems in Trigonometry

TOPIC 1: Basics of Trigonometry

Definitions

• Angles: Measured in degrees or radians.
• Line, Ray: Basic geometric concepts with defined endpoints and directions.
• Vertex: The corner point of an angle.
• Terminal Side: The position of the angle after it is opened from the initial side.

Angles Representation

• Classification of angles by their terminal sides:
  • Degrees vs Radians:
  • Conversion of degrees to radians: $ext{Radians} = \frac{ ext{Degrees}}{180^ ext{o}} imes ext{π}$
• Reference Angle: The acute angle formed between the terminal side of the angle and the horizontal axis.
• Example: The reference angle for 290º is calculated as:
  • $ext{Reference Angle} = 360^ ext{o} - 290^ ext{o} = 70^ ext{o}$

Geometry of Circles

Arc Length and Area of a Sector

• Arc Length formula: $s = r heta$, where $r$ is the radius and $heta$ is the central angle in radians.
• Area of a Sector: Given by the formula:
  $K = \frac{1}{2} r^2 heta$, where $θ$ is in radians.

Example Calculation

• Given a circle with radius 18 cm and a central angle of 50 degrees, we convert: $50^ ext{o} = \frac{5 ext{π}}{18}$ radians and calculate area:
  $K = \frac{1}{2} imes 18^2 imes \frac{5 ext{π}}{18} = 45 ext{π} ext{ cm}² ext{ or } 141 ext{ cm}² (rounded)$

Linear and Angular Velocity

• Linear Velocity (v): $v = \frac{ ext{arc length}}{ ext{time}}$
• Angular Velocity (ω): $ω = \frac{ ext{central angle in radians}}{ ext{time}}$
• Relationship: $v = rω$

TOPIC 2: Graphs of Trigonometric Functions

Graph Functions

1. Sin and Cos Functions:
   • General representations for sine and cosine functions:
     • $y = ext{sin}(x)$
     • $y = ext{cos}(x)$
2. Key properties include:
   • Periodicity, symmetry, and specific values at key angles (e.g., 0, 90, 180 degrees).

Domain and Range of Functions

• Sine and Cosine:
  • Domain: All Real Numbers
  • Range: $[-1, 1]$
• Cosecant and Secant:
  • Domain excludes points where sine or cosine is zero.
  • Their ranges are $(- ext{∞}, -1] ext{ or } [1, ext{∞})$

TOPIC 3: Angles of Elevation and Depression

Definition

• Angle of Elevation: The angle above horizontal (looking up to an object).
• Angle of Depression: The angle below horizontal (looking down to an object).

Example Problem

• A tree of height 100 ft casts a shadow of 120 ft.
  • To find the angle of elevation:
    $an(A) = \frac{ ext{opposite}}{ ext{adjacent}} = \frac{100}{120}$
  • Resulting in $A ≈ 40^ ext{o}$.

TOPIC 4: Basic Trigonometric Identities

Fundamental Identities

• Pythagorean Identity:
  • $ext{sin}^2(x) + ext{cos}^2(x) = 1$
• Reciprocal Relationships:
  • $ext{csc}(x) = \frac{1}{ ext{sin}(x)}$
  • $ext{sec}(x) = \frac{1}{ ext{cos}(x)}$
  • $ext{cot}(x) = \frac{1}{ ext{tan}(x)}$

TOPIC 5: Triangle Complexities

Laws of Sine and Cosine

1. Sine Law:
   $\frac{a}{ ext{sin}(A)} = \frac{b}{ ext{sin}(B)} = \frac{c}{ ext{sin}(C)}$
2. Cosine Law:
   $c^2 = a^2 + b^2 - 2ab ext{cos}(C)$

Area of Triangle

• Area can be calculated using:
  $ext{Area} = \frac{1}{2} imes ext{base} imes ext{height}$
  or using Heron's formula for given side lengths.

TOPIC 6: Trigonometric Equations and Inverses

Solving Equations

• Several tips include finding common angles, using identities, and ensuring to factor equations appropriately.
  • Example: Solving
    $ext{sin}(x) - 2 ext{sin}(x) ext{cos}(x) = 0$

Inverses

• Basic concept of inverses in trigonometry for arcsine, arccosine, etc.
  • Fundamental property: For all $y$ in Range of Function, if $y = ext{sin}(x), ext{then } x = ext{arcsin}(y)$

Solved Problems

1. Clock Minute Hand Movement:
   • Distance hand moves is calculated and results in approximately 25.1 cm.
2. Resolving angles and related calculations using standard trigonometric values in degrees and radians.

Conclusion

• Acknowledgment of completion and support available through UPLB DOST Scholars' Society for further questions and clarifications.