Scale Drawings, Bearings and Trigonometry

1. Scale Drawings

Scale drawings represent real-life objects or areas at a reduced size while maintaining proportions.

Scale Factors

A scale is written as $\text{Drawing Distance} : \text{Real Distance}$ (e.g., $1 : 50,000$).

  • Finding Real Distance: $\text{Real Distance} = \text{Drawing Distance} \times \text{Scale Factor}$.
  • Finding Drawing Distance: $\text{Drawing Distance} = \text{Real Distance} \div \text{Scale Factor}$.

2. Bearings

Bearings are used for navigation to specify direction.

Three-Figure Bearings

  • Measured from North.
  • Measured Clockwise.
  • Always written as three digits (e.g., $045^\circ$ instead of $45^\circ$).

Bearing Diagram

Reverse Bearings: To find the bearing from B back to A:

  • If bearing $A \rightarrow B$ is $\theta$, then bearing $B \rightarrow A = \theta + 180^\circ$ (if $\theta < 180^\circ$) or $\theta - 180^\circ$ (if $\theta \ge 180^\circ$).

3. Right-Angle Trigonometry

Applied to triangles with a $90^\circ$ angle.

SOH CAH TOA

  • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
  • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Solving for Sides and Angles

  • Sides: Use the ratio $\times$ the known side.
  • Angles: Use the inverse function ($\sin^{-1}, \cos^{-1}, \tan^{-1}$).

Sin Cosine and Tan

4. Exact Trigonometric Values

Certain angles have exact ratios that should be memorized.

$\theta$ $0^\circ$ $30^\circ$ $45^\circ$ $60^\circ$ $90^\circ$
$\sin \theta$ $0$ $1/2$ $1/\sqrt{2}$ $\sqrt{3}/2$ $1$
$\cos \theta$ $1$ $\sqrt{3}/2$ $1/\sqrt{2}$ $1/2$ $0$
$\tan \theta$ $0$ $1/\sqrt{3}$ $1$ $\sqrt{3}$ Undef

5. Non-Right-Angle Trigonometry

Applied to any triangle $ABC$ with sides $a, b, c$ opposite to angles $A, B, C$.

Sine Rule

Used when we have a “matching pair” (angle and opposite side). $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Cosine Rule

Used when we have SAS (Side-Angle-Side) or SSS (Side-Side-Side).

  • Finding Side: $a^2 = b^2 + c^2 - 2bc \cos A$
  • Finding Angle: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Area of a Triangle

$$\text{Area} = \frac{1}{2}ab \sin C$$

6. Advanced Trigonometry

Angles $> 90^\circ$

When angles exceed $90^\circ$, trigonometric values repeat or change signs based on the quadrant of the angle.

Trigonometric Graphs The periodic nature of trig functions means that different angles can produce the same value:

  • Sine Graph: A wave with a period of $360^\circ$. It is symmetric around $90^\circ$, meaning $\sin(\theta) = \sin(180^\circ - \theta)$.
  • Cosine Graph: A wave with a period of $360^\circ$. It is symmetric around $0^\circ/360^\circ$, meaning $\cos(\theta) = \cos(360^\circ - \theta)$.
  • Tangent Graph: A repeating curve with a period of $180^\circ$ and asymptotes at $90^\circ$ and $270^\circ$. It satisfies $\tan(\theta) = \tan(180^\circ + \theta)$.

Trigonometric Graphs

Inverse Values and Multiple Angles Because these functions are periodic, the inverse of a value often leads to more than one possible angle within $0^\circ$ to $360^\circ$.

  • The Calculator Limitation: Calculators are programmed to provide only the principal value (the angle closest to $0$). For example, $\sin^{-1}(0.5)$ will always return $30^\circ$.
  • Finding the Second Angle:
    • For Sine: If the calculator gives $\theta$, the other possibility is $180^\circ - \theta$.
    • For Cosine: If the calculator gives $\theta$, the other possibility is $360^\circ - \theta$.
    • For Tangent: If the calculator gives $\theta$, the other possibility is $180^\circ + \theta$.

This is why, in a triangle, you must check if an obtuse angle ($> 90^\circ$) is possible based on the given side lengths.

Graphical Explanation: Trigonometric circle Trigonometric circle values

3D Trigonometry

  • Definition: Finding lengths, angles, and distances within three-dimensional objects (like pyramids or prisms).
  • Strategy:
    • Step 1: Visualize and Sketch. Draw the 3D object and highlight the specific 2D triangles you need.
    • Step 2: Isolate 2D Triangles. Find a right-angled triangle that connects the known values to the unknown value.
    • Step 3: Chain of Calculation. Often, you must solve for an intermediate side length first (e.g., find the diagonal of the base) before you can find the final target value.
    • Step 4: Apply SOH CAH TOA or Pythagoras. Use the relevant rules sequentially.

3D Trig Example