IB Mathematics AI – Topic 3

The Three Basic Ratios:

Sine (sin):

\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]

Cosine (cos):

\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]

Tangent (tan):

\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]

Memory Aid - SOH CAH TOA:

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent

Pythagorean Theorem:

\[ a^2 + b^2 = c^2 \]

where c is the hypotenuse

⚠️ Common Pitfalls & Tips:

• Identify which side is opposite/adjacent to the given angle
• Hypotenuse is ALWAYS the longest side (opposite the right angle)
• Check calculator is in correct mode (degrees or radians)
• Use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles

Sine Rule Formula:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

Or equivalently:

\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \]

where a, b, c are sides opposite to angles A, B, C respectively

When to Use Sine Rule:

• When you know: AAS (two angles and one side)
• When you know: ASA (angle-side-angle)
• When you know: SSA (two sides and non-included angle) - ambiguous case

Finding a Side:

Use the form: \(\frac{a}{\sin A} = \frac{b}{\sin B}\)

Finding an Angle:

Use the form: \(\frac{\sin A}{a} = \frac{\sin B}{b}\)

⚠️ Common Pitfalls & Tips:

• Ambiguous case (SSA): May have 0, 1, or 2 solutions
• Label triangle clearly: side a opposite angle A, etc.
• Use GDC to avoid calculation errors
• Check answer is reasonable (angles sum to 180°)

Solution:

Given: A = 40°, B = 65°, b = 12 cm

Need to find: a

Step 1: Apply Sine Rule

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

Step 2: Substitute values

\[ \frac{a}{\sin 40°} = \frac{12}{\sin 65°} \]

Step 3: Solve for a

\[ a = \frac{12 \times \sin 40°}{\sin 65°} \]

\[ a \approx \frac{12 \times 0.6428}{0.9063} \approx 8.51 \text{ cm} \]

Answer: a = 8.51 cm (3 s.f.)

Cosine Rule Formula (for finding a side):

\[ c^2 = a^2 + b^2 - 2ab\cos C \]

Or equivalently for other sides:

\[ a^2 = b^2 + c^2 - 2bc\cos A \]

\[ b^2 = a^2 + c^2 - 2ac\cos B \]

Cosine Rule (for finding an angle):

\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

When to Use Cosine Rule:

• When you know: SAS (two sides and included angle)
• When you know: SSS (all three sides)
• When sine rule cannot be used

⚠️ Common Pitfalls & Tips:

• The angle in the formula is BETWEEN the two known sides
• Don't forget the -2ab cos C term (not +)
• When finding angle, use cos⁻¹ function on GDC
• Check calculator mode (degrees or radians)

1. Basic Formula (with base and height):

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

2. Using Sine (two sides and included angle):

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

where a and b are two sides, C is the included angle

Most commonly used in IB exams!

3. Heron's Formula (all three sides known):

\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]

where \(s = \frac{a+b+c}{2}\) is the semi-perimeter

⚠️ Common Pitfalls & Tips:

• For \(\frac{1}{2}ab\sin C\), angle C must be BETWEEN sides a and b
• Don't forget the 1/2 factor
• Area is always in square units (cm², m²)
• Use GDC to calculate - faster and more accurate

Solution:

Given: Two sides and included angle (SAS)

PQ = 8 cm, PR = 10 cm, angle P = 50°

Step 1: Choose appropriate formula

Use: \(\text{Area} = \frac{1}{2}ab\sin C\)

Step 2: Substitute values

\[ \text{Area} = \frac{1}{2} \times 8 \times 10 \times \sin 50° \]

Step 3: Calculate

\[ \text{Area} = \frac{1}{2} \times 80 \times 0.7660 \]

\[ \text{Area} \approx 40 \times 0.7660 = 30.6 \text{ cm}^2 \]

Answer: 30.6 cm²

Conversion Between Degrees and Radians:

\[ 180° = \pi \text{ radians} \]

\[ 360° = 2\pi \text{ radians} \]

Conversion Formulas:

Degrees to Radians:

\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]

Radians to Degrees:

\[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \]

Common Radian Values:

• 30° = \(\frac{\pi}{6}\) rad
• 45° = \(\frac{\pi}{4}\) rad
• 60° = \(\frac{\pi}{3}\) rad
• 90° = \(\frac{\pi}{2}\) rad
• 180° = \(\pi\) rad

⚠️ Common Pitfalls & Tips:

• CRITICAL: Check calculator mode matches question units
• Radians don't have a degree symbol (°)
• Use π button on calculator, not 3.14
• Arc length and sector area formulas REQUIRE radians

Definitions:

• Arc: Part of the circumference of a circle
• Sector: Region bounded by two radii and an arc (like a pizza slice)
• Segment: Region bounded by a chord and an arc

Arc Length (θ in radians):

\[ l = r\theta \]

where r = radius, θ = angle in radians

Arc Length (θ in degrees):

\[ l = \frac{\theta}{360} \times 2\pi r \]

Sector Area (θ in radians):

\[ A = \frac{1}{2}r^2\theta \]

Sector Area (θ in degrees):

\[ A = \frac{\theta}{360} \times \pi r^2 \]

Segment Area:

\[ A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} \]

⚠️ Common Pitfalls & Tips:

• Must use radians for formulas \(l = r\theta\) and \(A = \frac{1}{2}r^2\theta\)
• Convert degrees to radians first if needed
• Arc length has same units as radius (cm, m)
• Sector area has squared units (cm², m²)

Triangle Type

• Right angle? Use SOH CAH TOA
• No right angle? Use sine or cosine rule

Which Rule?

• AAS/ASA: Sine rule
• SAS/SSS: Cosine rule

Circles

• Convert to radians first!
• Arc: \(l = r\theta\)
• Sector: \(A = \frac{1}{2}r^2\theta\)

Calculator Mode

• Always check mode!
• Degrees (°) or Radians (rad)
• Match question units