Mathematics 2: Trigonometry and Geometric Properties

Angle Measurement and Conversion

• Angles are measured in degrees or radians, where a full circle is $360^{\circ}$ or $2\pi$ rad.
• Conversion formulas:
  • $1^{\circ} = \frac{\pi}{180}$
  • Convert radians to degrees: Multiply the value by $\frac{360}{2\pi}$.
  • Example: $\frac{\pi}{2} = 90^{\circ}$.
  • Example: $120^{\circ} = \frac{2}{3}\pi$.

Trigonometric Ratios and Right-Angled Triangles

• Basic ratios based on the Hypotenuse, Adjacent, and Opposite sides:
  • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
  • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
  • $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\text{Opposite}}{\text{Adjacent}}$
• Pythagoras’s Theorem: Applicable only to right-angled triangles:
  • $b^2 = a^2 + c^2$ (where $b$ is the Hypotenuse).
• Solving a right-angled triangle involves finding all side lengths and angles using these ratios and the property that the two non-right angles sum to $\frac{\pi}{2}$.

Properties of Trigonometric Functions

• Sine and Cosine:
  • Continuous graphs oscillating between $-1$ and $1$.
  • Standard period is $2\pi$.
  • Cosine is sine displaced by $\frac{\pi}{2}$, so $\cos(x - \frac{\pi}{2}) = \sin(x)$.
  • Odd/Even: $\sin(-x) = -\sin(x)$ (odd); $\cos(-x) = \cos(x)$ (even).
• Tangent:
  • Discontinuous at multiples of $\frac{\pi}{2}$.
  • Monotonically increasing between discontinuities.
  • Standard period is $\pi$.
• Amplitude and Phase:
  • Amplitude (peak/maximum value) is adjusted by multiplying the function by a scalar (e.g., $3\sin(x)$).
  • Phase is adjusted by shifting along the x-axis (e.g., $\cos(x - \frac{\pi}{4})$).
• Periodicity Changes: Multiplication inside the argument changes the period. For example, $\sin(4x)$ has a period of $\frac{\pi}{2}$, and $\sin(\frac{1}{2}x)$ has a period of $4\pi$.

Trigonometric Identities and Inverse Functions

• Pythagorean Identity: $\cos^2(\theta) + \sin^2(\theta) = 1$
• Compound Angle Identities:
  • $\sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)$
  • $\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)$
• Inverse Functions: The functions $\text{arcsin}(x)$, $\text{arccos}(x)$, and $\text{arctan}(x)$ are used to determine the angle $\theta$ from a known ratio.

Solving General Triangles

• Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
• Cosine Rule: used for non-right triangles:
  • $a^2 = b^2 + c^2 - 2bc\cos(A)$
  • $b^2 = a^2 + c^2 - 2ac\cos(B)$
  • $c^2 = a^2 + b^2 - 2ab\cos(C)$
• Area of General Triangles:
  • Trigonometric Area: $\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2}ac\sin(B) = \frac{1}{2}bc\sin(A)$
  • Heron’s Formula: $\text{Area} = \sqrt{|s(s - a)(s - b)(s - c)|}$, where $s = \frac{a + b + c}{2}$ (semi-perimeter).

Perimeter and Area of Common Shapes

• Rectangle: $\text{Area} = a \cdot b$; $\text{Perimeter} = 2a + 2b$
• Parallelogram: $\text{Area} = h \cdot b$; $\text{Perimeter} = 2a + 2b$
• Trapezium: $\text{Area} = h \cdot \frac{a + b}{2}$; $\text{Perimeter} = 2a + 2b$
• Circle: $\text{Area} = \pi r^2$; $\text{Circumference} = 2\pi r$
• Circle Sector: $\text{Area} = \frac{\theta^{\circ}}{360^{\circ}}\pi r^2$
• Arc Length: $s = \theta r$ (for $\theta$ in radians).

Volume and Surface Area of 3D Objects

• Cuboid: $\text{Volume} = l \times w \times h$; $\text{Surface Area} = 2(lw + lh + wh)$.
• Cylinder: $\text{Volume} = \pi r^2 h$; $\text{Total Surface Area} = 2\pi rh + 2\pi r^2$.
• Pyramids and Cones:
  • $\text{Volume} = \frac{1}{3} \times \text{Area of Base} \times \text{Perpendicular Height}$.
  • Cone Surface Area: $\pi rl + \pi r^2$, where the slant height $l = \sqrt{r^2 + h^2}$.
• Sphere:
  • $\text{Volume} = \frac{4}{3}\pi r^3$
  • $\text{Surface Area} = 4\pi r^2$