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.
     • $\tan(15^{\circ}) = \frac{H<em>1}{d} \implies H</em>1 = d \times \tan(15^{\circ})$
     • $\tan(2^{\circ}) = \frac{H<em>2}{d} \implies H</em>2 = d \times \tan(2^{\circ})$
  3. The combined height is: $H<em>1 + H</em>2 = 200$
  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.