Trigonometry

A 2 m stick casts a 1.5 m shadow at the same time a building casts a 18 m shadow. The sun's rays form equal angles with the ground for both, creating two similar right triangles (AA similarity — both share a right angle and equal sun-angle).

Surveyors, architects, and engineers use this principle constantly — measuring inaccessible heights via proportional shadows or mirror reflections on the ground.

ADA guidelines require wheelchair ramps to have a maximum slope of 1:12 (rise:run). What is the ramp angle?

A building entrance 0.6 m above ground needs a ramp at least $0.6 \times 12 = 7.2$ m long horizontally. The actual ramp length (hypotenuse) is $\frac{0.6}{\sin 4.76°} \approx 7.22$ m. Civil engineers use SOHCAHTOA every day for grading roads, designing ramps, and calculating sight lines.

A surveyor stands at point A and measures distances to two boundary markers: AB = 340 m and AC = 510 m. The angle between the two sight lines is 72°. What is the distance BC between the markers?

Sine Law and Cosine Law are the backbone of triangulation — used in GPS, land surveying, marine navigation, and satellite ranging to compute distances and positions when direct measurement is impossible.

A house has a roof with a 12/12 pitch (rises 12 inches for every 12 inches of run). The roof angle is $\arctan(12/12) = 45°$. For a house 10 m wide, each rafter spans half the width (5 m run), so the rafter length is:

Carpenters and structural engineers memorise the special triangle ratios — a 30° roof (low slope, common in warm climates) versus a 60° roof (steep, sheds snow) have rafter lengths directly derived from these exact values.

A Ferris wheel has radius 25 m and its centre is 27 m above the ground. A rider boards at the bottom (angle = 270° on the unit circle, or $-\pi/2$). After rotating through angle $\theta$ from the 3-o'clock position, the rider's height is:

At the top ($\theta = 90°$): $h = 27 + 25(1) = 52$ m. At the bottom ($\theta = 270°$): $h = 27 + 25(-1) = 2$ m. Theme park engineers use unit-circle relationships to model rider position, speed, and g-forces at every point of the rotation.

A 500 N crate sits on a conveyor belt inclined at 150° to the positive x-axis (measuring counter-clockwise). The horizontal and vertical force components are:

Mechanical engineers decompose forces at arbitrary angles daily — designing conveyor systems, crane booms, and bridge trusses. Knowing how to handle angles beyond 90° via reference angles is essential for resolving vectors in any quadrant.

A harbour's tide varies between 1.2 m (low) and 5.8 m (high) with a period of approximately 12.4 hours. If high tide occurs at $t = 0$:

Marine engineers, harbour pilots, and coastal planners use sinusoidal models to predict water levels for docking schedules, flood barriers, and coastal construction timing.

A car travels at 90 km/h. Its tyres have radius 0.32 m. The angular velocity $\omega$ (in rad/s) of each wheel is:

Radians are the natural unit for rotational mechanics. Every rotating machine — engines, turbines, hard drives, centrifuges — is analysed using radians because the arc-length formula $s = r\theta$ and the velocity formula $v = r\omega$ only work directly in radians.

In electrical engineering, instantaneous power in an AC circuit is:

Using the product-to-sum identity (derived from the Pythagorean and angle-sum identities), this simplifies to:

The first term gives the average (real) power; the second is the oscillating component. Without trig identities, analysing AC circuits, signal processing, and electromagnetic wave interference would be impossibly complex.

A pendulum's angular displacement is modelled by $\theta(t) = 0.15 \sin(2\pi t)$ (radians), where $t$ is in seconds. A sensor triggers when $\theta = 0.075$ rad. When does it trigger in the first second?

Robotics engineers solving for precise timing of periodic events — oscillating arms, vibrating components, rotating gears — solve trig equations like this constantly to synchronise sensors, actuators, and control loops.