Accuracy, Geometry, and Trigonometry Study Guide

Representing Numbers: Accuracy and Geometry

• Significant Figures and Rounding Rules:

  • Significant figures (digits) are the digits in a number that can be determined accurately.
  • Example: A scale registering grams to the hundredths place until $99.99\,g$ can measure up to 4 significant figures of accuracy.
  • In calculations, round final answers to the same number of significant figures as the least accurate measurement used in the operation.
  • Exam Instruction: Final answers should be written as exact values or rounded to at least 3 significant figures unless otherwise specified. Intermediate calculations must not be rounded; only the final result is rounded.

• Bounds and Error Quantification:

  • Measurements are often approximate. If a measurement $M$ is accurate to a specific unit $u$, the exact value $V$ lies within the interval:     $M - 0.5u \le V < M + 0.5u$
  • The endpoints of this defined interval are the lower and upper bounds.
  • Example: A bag of coffee weighing $541.5\,g$ (accurate to the nearest $0.1\,g$) has an exact weight $w$ in the interval $541.45\,g \le w < 541.55\,g$.

• Measurement and Percentage Error:

  • Measurement Error: The difference between the approximate value ($V_A$) and the exact value ($V_E$).     $\text{Measurement error} = V_A - V_E$
  • Percentage Error Formula: To find error as a percentage of the exact value, the formula is:     $\text{Percentage error} = \left| \frac{V_A - V_E}{V_E} \right| \times 100\%$
  • Note: The absolute value bars ensures the resulting error percentage is a positive value.

Exponents and Standard Form

• Standard Form (Scientific Notation):

  • Standard form is written as $a \times 10^k$, where the coefficient $a$ satisfies $1 \le a < 10$ and the exponent $k \in \mathbb{Z}$.
  • This notation is used for efficiency with very large (astronomy, macroeconomics) or very small (chemistry) numbers.

• Operations with Standard Form:

  • Multiplication: $(b \times 10^m) \times (c \times 10^n) = (b \times c) \times 10^{m+n}$.
  • Division: $\frac{b \times 10^m}{c \times 10^n} = \frac{b}{c} \times 10^{m-n}$.

• Exponent Rules for $a > 0$ and $m, n \in \mathbb{Q}$:

  • Product Rule: $a^m \times a^n = a^{m+n}$.
  • Power of a Power: $(a^m)^n = a^{mn}$.
  • Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$.
  • Negative Exponents: $x^{-n} = \frac{1}{x^n}$.
  • Rational Exponents: $x^{\frac{p}{q}} = \sqrt[q]{x^p} = (\sqrt[q]{x})^p$.
  • Square Root Notation: $x^{\frac{1}{2}} = \sqrt{x}$.

Right-Angled Trigonometry and Bearings

• Trigonometric Ratios in Right Triangles:

  • $\sin(\hat{A}) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
  • $\cos(\hat{A}) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • $\tan(\hat{A}) = \frac{\text{Opposite}}{\text{Adjacent}}$
  • The mnemonic SOH-CAH-TOA is used to remember these definitions.

• Angles of Elevation and Depression:

  • Angle of Elevation: The angle formed between a horizontal line at eye level and the line of sight when looking up at an object.
  • Angle of Depression: The angle formed between the horizontal line at eye level and the line of sight when looking down at an object.

• Bearings:

  • A bearing is an angle measured clockwise from North ($000^{\circ}$ or $360^{\circ}$).
  • East is $090^{\circ}$, South is $180^{\circ}$, and West is $270^{\circ}$.

Non-Right Triangle Geometry

• The Sine Rule:

  • Applicable to any triangle (right or non-right).
  • $\frac{a}{\sin(\hat{A})} = \frac{b}{\sin(\hat{B})} = \frac{c}{\sin(\hat{C})}$
  • Or, to solve for an angle: $\frac{\sin(\hat{A})}{a} = \frac{\sin(\hat{B})}{b} = \frac{\sin(\hat{C})}{c}$.

• The Ambiguous Case of the Sine Rule:

  • Occurs when two sides and one non-included angle are known, and the unknown angle is opposite the longer of the two known sides.
  • This scenario may result in two possible triangles: one acute and one obtuse.
  • The two possible solutions for the angle are supplementary (sum to $180^{\circ}$).

• Area of Any Triangle:

  • The area is calculated using two sides and the included angle:     $\text{Area} = \frac{1}{2}ab\sin(\hat{C})$

• The Cosine Rule:

  • Finding a side: $a^2 = b^2 + c^2 - 2bc\cos(\hat{A})$.
  • Finding an angle: $\cos(\hat{A}) = \frac{b^2 + c^2 - a^2}{2bc}$.
  • Use the Cosine Rule when all three sides are known or when two sides and the included angle are known.

Circular and Three-Dimensional Geometry

• Arc Length and Sector Area (Central angle $\theta$ in degrees):

  • Length of an Arc: $\frac{\theta}{360} \times 2\pi r$
  • Area of a Sector: $\frac{\theta}{360} \times \pi r^2$
  • A segment's area is found by subtracting the triangle area from the sector area ($\text{Sector} - \text{Triangle}$).

• Volumes of 3D Solids:

  • Prism/Cylinder: $V = \text{Base Area} \times h$
  • Pyramid/Cone: $V = \frac{1}{3} \times \text{Base Area} \times h$
  • Sphere: $V = \frac{4}{3}\pi r^3$

• Surface Area of 3D Solids:

  • Lateral Area (LA): The area of the faces excluding the base(s).
  • Prisms/Cylinders: $SA = LA + 2B$. For cylinders: $SA = 2\pi rh + 2\pi r^2$.
  • Regular Pyramids: $LA = \frac{1}{2}pl$, where $p$ is base perimeter and $l$ is slant height.
  • Cones: $LA = \pi rl$ ($l = \text{slant height}$). Total $SA = \pi rl + \pi r^2$.
  • Sphere: $SA = 4\pi r^2$.

• Angle Between a Line and a Plane:

  • For line $AG$ and plane $ABCD$: Drop a perpendicular from $G$ to the plane reaching point $C$; the angle is $\angle GAC$, forming right-angled triangle $\Delta GAC$.

Examples and Applications

• Mount Everest Surveying: Andrew Waugh announced the height as $8840\,m$ in 1856 based on the Great Trigonometric Survey (theodolite measurements). Recent surveys adjust this to $8848\,m$.
• NASA Apollo Software: Lead developer Margaret Hamilton produced stacks of navigation code books approximate to her own height in 1969.
• Distance and Speed: Light traveling from the Sun to Earth ($1.50 \times 10^{11}\,m$) at $3 \times 10^8\,m/s$ takes approximately $500\,s$.
• Bicycle Statistics: Japan has approximately $72$ million bicycles (nearest million). Average daily travel per person is $2\,km$ (nearest $km$).

Questions & Discussion

• Investigation 1 (Hamilton NASA):
  • 1. Estimate the height of the stack of code books next to Margaret Hamilton. What assumptions are made?
  • 2. Estimate the number of pages in the code. How would you do this?
  • 3. Factual: What is an estimate vs estimation? Conceptual: Why are estimations useful?
• Investigation 2 (Measurement Error):
  • Tomi and Massimo measure a yard ($92.44\,cm$) and a foot ($31.48\,cm$). The exact values are $91.44\,cm$ and $30.48\,cm$.
  • Comparison of inaccuracy: Is measurement error more useful as a raw value or a percentage of the total length?
• Investigation 5 (Ambiguous Case):
  • 1. Use the Sine Rule to find $\hat{D}$ in a triangle with $B=19^{\circ}$, $c=9.2\,cm$, and $a=3.7\,cm$. Explain why the solution might be inconsistent with a diagram.
  • 2. If one solution is $39^{\circ}$, the other solution in the ambiguous case is $180 - 39 = 141^{\circ}$.
• TOK Considerations:
  • How does the perception of language distort understanding of measurements?
  • If DNA or drug testing has a percentage error of $0.0526$, can a person be certain of guilt? What is an acceptable error rate for high stakes?
  • What are the ethical implications of rounding numbers?