Principles of Trigonometry, Pythagoras, and Bearings

Pythagoras' Theorem describes the relationship in a right triangle:
$a^2 + b^2 = c^2$. The hypotenuse ($c$) is opposite the right angle.
Example: In a triangle with a hypotenuse of $5.6 \, m$ and one side of $2.84 \, m$, calculate the missing side using $\text{missing side} = \\sqrt{5.6^2 - 2.84^2}$.
In 3D, such as a cube with side lengths of $10$, find a diagonal (e.g., $AG$) by first calculating the base diagonal with $10^2 + 10^2 = \text{diagonal}^2$, then using that diagonal as a side in a second triangle to find lengths like approximately $17$.
Similarly, find perpendicular heights in pyramids by identifying right triangles and using slant heights.

Trigonometric Ratios and Conversions

Trigonometric ratios define relationships between angles and sides:

• Hypotenuse ($H$) is opposite right angle, opposite ($O$) is across from reference angle, adjacent ($A$) is next to it.

• Ratios:
  
  

  • $\text{sin}(\theta) = \frac{O}{H}$

  • $\text{cos}(\theta) = \frac{A}{H}$

  • $\text{tan}(\theta) = \frac{O}{A}$
    To find unknown angles, apply inverse functions (e.g., $A = \text{sin}^{-1}(\frac{35}{75})$).
    Relationships: $\frac{\text{sin}(x)}{\text{cos}(x)} = \text{tan}(x)$ and $\text{sin}(x) = \text{cos}(90^{\text{◦}} - x)$.

Angle Measurement

Angles measured in degrees, minutes, and seconds can be converted into decimal degrees. For example, the angle $180^{\text{◦}} 52' 34''$ calculates as:
$180 + \frac{52}{60} + \frac{34}{3600}$.

Angles of Elevation and Depression

• Angle of Elevation: Measured from horizontal eye level upwards. For instance, to find the height $x$ of a tree from $23 \, m$ away at an elevation angle of $35^{\text{◦}}$:
  $\tan(35^{\text{◦}}) = \frac{x}{23}$. If the observer is $1.5 \, m$ tall, add this height to $x$ for total height.

• Angle of Depression: Measured down from horizontal eye level; it’s equal to the angle of elevation from the object looking back up. For example, to find distance $x$ from a height of $2.8 \, m$ with an angle of depression:
  $\tan(47^{\text{◦}}) = \frac{28.8}{x}$.

Navigation and Bearings

• Bearings: Directions as angles; measured clockwise from North. True bearings are expressed in three digits followed by $T$ (e.g., $030^{\text{◦}}T$).
  To find a true bearing, center a compass on a point and measure clockwise.

• Compass Bearings: Start with North/South, followed by angle toward East/West (e.g., $S30^{\text{◦}}E$).
  Converting between true and compass bearings involves identifying the travel quadrant and angle from North/South.

Questions & Discussion

During the review, a clarification arose about including the height of the observer in the tree problem. The total height incorporates the calculated triangle height plus observer height (e.g., $16.10 \, m + 1.5 \, m = 17.60 \, m$).
In navigation, an exchange about quadrant numbering reinforced that while standard coordinate geometry employs specific orders, bearings focus on clockwise movement from North through East, South, and West.
A bearing of $147^{\text{◦}}T$ lies between East and South, precisely $57^{\text{◦}}$ past East.
Additionally, the relationship between sine and cosine for complementary angles was explained, showing that $\text{sin}(35^{\text{◦}})$ equals $\text{cos}(55^{\text{◦}})$ based on angle sums to $90^{\text{◦}}$.