Trigonometry Notes

Foundations and Applications of Trigonometry

• Trigonometry originates from the Greek terms trigonon (triangle) and metron (measure).

• Its primary function is calculating unknown lengths and angles in triangles that are not accessible via physical measurement.

• Key fields of application include engineering, navigation, surveying, astronomy, electronics, and construction.

Essential Terminology

• Bearing: An angular measurement indicating direction from a specific reference point.

• Complementary Angles: Any two angles that sum to exactly $90^{\circ}$.

• Included Angle: The specific angle located between two known sides of a triangle.

• Exact Ratio: Expressing the sine, cosine, or tangent of special angles ($30^{\circ}$, $45^{\circ}$, $60^{\circ}$) as surds or fractions rather than decimal approximations.

Right-Angled Trigonometry

• The three fundamental trigonometric ratios are defined based on a right-angled triangle with an angle $\theta$:

  • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Procedures for Calculating Sides:

  • Identify the two sides involved (labeled with respect to the known angle).

  • Select the ratio (sin, cos, or tan) connecting those sides.

  • Solve the resulting equation for the unknown side.

  • Example: For a triangle with hypotenuse $15.6\,m$ and angle $35^{\circ}26'$ finding the adjacent side $w$: $\cos(35^{\circ}26') = \frac{w}{15.6}$, leading to $w = 15.6 \times \cos(35^{\circ}26') \approx 12.7\,m$.

• Procedures for Calculating Angles:

  • Set up the ratio using the two known sides.

  • Use the inverse trigonometric function (e.g., $\tan^{-1}(x)$) to find the angle.

  • Example: For opposite side $35\,mm$ and adjacent side $24\,mm$, $\tan(\theta) = \frac{35}{24}$. Solving gives $\theta \approx 55^{\circ}34'$.

Angles of Elevation and Depression

• Angle of Elevation: The angle measured upwards from the horizontal line to the line of sight toward an object.

• Angle of Depression: The angle measured downwards from the horizontal line to the line of sight toward an object.

• By the principle of alternate angles on parallel horizontal lines, the angle of elevation from a target is equal to the angle of depression from the observer.

• These problems frequently utilize the tangent ratio.

• Example calculation: From a yacht $190\,m$ from a cliff base, an angle of elevation of $18^{\circ}$ yields a cliff height $x$ via $\tan(18^{\circ}) = \frac{x}{190}$, resulting in $x \approx 61.7\,m$.

Bearings and Navigation

• Three-Figure (True) Bearings:

  • Measured clockwise from North ($000^{\circ}$).

  • Expressed using three digits (e.g., $052^{\circ}$).

  • Range from $000^{\circ}$ to $360^{\circ}$.

• Compass Bearings:

  • Utilize sixteen points of the mariner's compass (N, NNE, NE, ENE, E, etc.).

  • Based on quadrants relative to North, South, East, and West.

• Reverse Bearings: To find the bearing of a starting point from a destination, the alternate angle theorem is often applied to the parallel North lines. For instance, if the bearing from town to plane is $122^{\circ}$, the bearing from plane back to town is $302^{\circ}$.

Pythagoras' Theorem and Trigonometry in 3D

• Solving three-dimensional problems often requires multiple steps involving right-angled triangles in different planes.

• Finding the Longest Diagonal of a Prism:

  1. Use Pythagoras' theorem ($a^2 + b^2 = c^2$) on the base to find the base diagonal.

  2. Use that base diagonal and the vertical height as sides of a second right-angled triangle to find the space diagonal.

  • Example: In a box $18\,cm \times 8\,cm \times 4\,cm$, the base diagonal is $\sqrt{18^2 + 8^2} = \sqrt{388} \approx 19.7\,cm$. The space diagonal is $\sqrt{(\sqrt{388})^2 + 4^2} = \sqrt{404} \approx 20.1\,cm$.

• 3D Angle Calculations: Angles of inclination are found by relating vertical heights to horizontal distances within the 3D structure.

Trigonometric Relations and Complementary Angles

• Complementary angles sum to $90^{\circ}$.

• Complementary Ratios:

  • $\sin(A) = \cos(90^{\circ} - A)$

  • $\cos(A) = \sin(90^{\circ} - A)$

• Example: $\sin(35^{\circ}) = \cos(55^{\circ})$.

Exact Trigonometric Ratios

Standard values for special angles are derived from isosceles right-angled triangles and equilateral triangles:

• For $45^{\circ}$:

  • $\sin(45^{\circ}) = \frac{1}{\sqrt{2}}$

  • $\cos(45^{\circ}) = \frac{1}{\sqrt{2}}$

  • $\tan(45^{\circ}) = 1$

• For $30^{\circ}$:

  • $\sin(30^{\circ}) = \frac{1}{2}$

  • $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$

  • $\tan(30^{\circ}) = \frac{1}{\sqrt{3}}$

• For $60^{\circ}$:

  • $\sin(60^{\circ}) = \frac{\sqrt{3}}{2}$

  • $\cos(60^{\circ}) = \frac{1}{2}$

  • $\tan(60^{\circ}) = \sqrt{3}$

Trigonometric Functions and the Unit Circle

• The Unit Circle: A circle with a radius ($r$) of $1$ centered at the origin $(0,0)$.

• For any point $P(x, y)$ on the circle at angle $\theta$:

  • $\cos(\theta) = x\text{-coordinate}$

  • $\sin(\theta) = y\text{-coordinate}$

  • $\tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}$

• Quadrant Signs:

  • 1st Quadrant ($0^{\circ}$–$90^{\circ}$): All (Sin, Cos, Tan) are positive.

  • 2nd Quadrant ($90^{\circ}$–$180^{\circ}$): Only Sin is positive. Cos and Tan are negative.

  • 3rd Quadrant ($180^{\circ}$–$270^{\circ}$): Only Tan is positive. Sin and Cos are negative.

  • 4th Quadrant ($270^{\circ}$–$360^{\circ}$): Only Cos is positive. Sin and Tan are negative.

• Supplementary Angle Relations (2nd Quadrant):

  • $\sin(180^{\circ} - A) = \sin(A)$

  • $\cos(180^{\circ} - A) = -\cos(A)$

  • $\tan(180^{\circ} - A) = -\tan(A)$

Trigonometric Graphs

• Sine Curve: A wave ranging between $-1$ and $1$, passing through $(0,0)$, $(90,1)$, $(180,0)$, $(270,-1)$, and $(360,0)$.

• Cosine Curve: A wave ranging between $-1$ and $1$, starting at $(0,1)$, passing through $(90,0)$, $(180,-1)$, $(270,0)$, and ending at $(360,1)$.

• Tangent Curve: Features a period of $180^{\circ}$ and vertical asymptotes at $90^{\circ}$ and $270^{\circ}$ where the function is undefined.

Trigonometric Equations

• Equations such as $\sin(\theta) = 0.7538$ can have multiple solutions within a $0^{\circ}$ to $180^{\circ}$ range.

• In this range, if $\sin(\theta)$ is positive, solutions exist in the 1st (acute) and 2nd (obtuse) quadrants.

• If $\cos(\theta)$ or $\tan(\theta)$ is negative, the solution will be obtuse (2nd quadrant).

• Example: $\tan(\theta) = -2.5$ results in an obtuse angle of approximately $112^{\circ}$.

The Sine Rule

• Applicable to any triangle (not just right-angled) to relate sides and opposite angles.

• Formula: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Conditions for Use: Finding a side when two angles and one side are known, or finding an angle when two sides and one opposite angle are known.

• The Ambiguous Case: When using the sine rule to find an angle, there may be two valid triangles (one acute, one obtuse) if the sum of the angles does not exceed $180^{\circ}$.

The Cosine Rule

• Relates three sides and one angle of any triangle.

• Finding a Side: $a^2 = b^2 + c^2 - 2bc \cos(A)$

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

• Conditions for Use: Finding the third side when two sides and the included angle are known (SAS), or finding any angle when three sides are known (SSS).

• If applied to a right-angled triangle ($A = 90^{\circ}$), the formula simplifies to Pythagoras' theorem ($a^2 = b^2 + c^2$) because $\cos(90^{\circ}) = 0$.

Area of a Triangle

• The trigonometric formula for area $(A)$ requires two sides and the included angle.

• Formula: $\text{Area} = \frac{1}{2}ab \sin(C)$

• This formula replaces the need for perpendicular height $(h)$ when the included angle is known.

• Heron's Formula (Alternative): Used when only three sides are known:

  • $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$

  • Where $s$ is the semi-perimeter: $s = \frac{a + b + c}{2}$.