Geometry and Trigonometry Study Notes

Geometry and Trigonometry in Right Angles

Introduction to Right Angles and Triangles

  • Right Angle Triangle Basics
    • A right triangle contains a 90-degree angle, referred to as angle C.
    • The sides opposite to angles are denoted by the same letter as the angle.
    • Angle A -> Side a
    • Angle B -> Side b
    • Angle C -> Side c (hypotenuse)

Finding Missing Angles and Sides

  1. Example of a Right Triangle
    • Given: Angle B = 30 degrees; Hypotenuse (c) = 2 units
    • Find: Angles A and C and lengths of sides a and b.
    • Angle A can be determined as 60 degrees (since 90 - 30 = 60).
    • Angle C remains 90 degrees as per the right triangle definition.

Determining Side Lengths Using Trigonometric Functions

Finding Side b

  • Relation: Sine Function
    • Sine of an angle = Opposite side / Hypotenuse
    • For angle B:
      \sin(30^{\circ}) = \frac{b}{2}
    • Solution Steps:
    1. Multiply both sides by 2:
      • b = 2 \times \sin(30^{\circ})
    2. Since \sin(30^{\circ}) = \frac{1}{2},
      • b = 2 \times \frac{1}{2} = 1.

Finding Side a

  • Relation: Cosine Function
    • Cosine of an angle = Adjacent side / Hypotenuse
    • For angle A:
      \cos(30^{\circ}) = \frac{a}{2}
    • Solution Steps:
    1. Multiply both sides by 2:
      • a = 2 \times \cos(30^{\circ})
    2. Utilization of unit circle coordinates:
      • \cos(30^{\circ}) = \frac{\sqrt{3}}{2}
      • a = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}.

Verification with the Pythagorean Theorem

  • Pythagorean Theorem: a^2 + b^2 = c^2
    • Check Values:
    • \sqrt{3}^2 + 1^2 = 2^2
    • Calculation:
      • 3 + 1 = 4
      • Confirms the measures are accurate.

Complex Applications with Angles of Elevation and Depression

Problem Example 15

  • Scenario: A 200-foot tall monument is viewed from a window.
    • Angle of Elevation to the top: 15 degrees.
    • Angle of Depression to the bottom: 2 degrees.
  • Objective: Find distance from the person to the monument.
  • Solution Steps:
    1. Create two triangles to represent the distances to top (H1) and bottom (H2) of the monument:
    2. Write equations for both triangles using trigonometric relationships.
      • \tan(15^{\circ}) = \frac{H1}{d} \implies H1 = d \times \tan(15^{\circ})
      • \tan(2^{\circ}) = \frac{H2}{d} \implies H2 = d \times \tan(2^{\circ})
    3. The combined height is: H1 + H2 = 200
    4. Final equation becomes:
      d \cdot \tan(15^{\circ}) + d \cdot \tan(2^{\circ}) = 200
    5. Factor out d: d = \frac{200}{\tan(15^{\circ}) + \tan(2^{\circ})}.

Numerical Example

  • Calculate d:
    1. d = \frac{200}{\tan(15^{\circ}) + \tan(2^{\circ})}
    2. Use calculator for evaluation.
      • Solution yields approximately 660 feet after rounding.

Further Applications and Problem 16

  • Next problem involves using applied knowledge to a similar problem with different heights and angles.
    • Emphasizes on step-by-step resolution and checking trig functions.

General Strategies for Trigonometry Problems

  • Always sketch the problem to visualize triangle(s).
  • Identify known values (sides and angles) and label each clearly.
  • Use the proper trigonometric function based on the required side or angle (Sine, Cosine, Tangent).
  • Factor equations cautiously when combining results from multiple triangles.

Additional Examples: Building Height and Lightning Rods

  1. Lightning Rod Problem

    • Analyzing height differences by calculating using angles of elevation from a distance to find total height reported as height of rod.
    • Ensure to clearly label known vs unknown heights and solve for the difference determining the total height.

  2. Each problem is approached using similar underlying principles: establishing the relationships between angles, distances, and respective functions to calculate unknown values.