Math for ME · Chapter 2 of 19 · Beginner
Trigonometry and Geometry for Mechanics
Force decomposition, moments, slopes, and circular motion all reduce to triangles. Master the triangle and half of Statics is pre-solved.
The thread: You can now rearrange any relationship. The next question is geometry: forces lean at angles and motion repeats, and both reduce to the triangle.
Readiness check
From Algebra, Functions, and Engineering Notation. Tick only what you can do closed-notes.
- Rearrange a formula for any variable with units intact.
- Use the Pythagorean theorem fluently.
- Operate a calculator in degree mode and radian mode, on purpose.
- Sketch a function from its equation (the Algebra and Functions shapes).
- Work with ratios and proportions.
The core idea
sin, cos, and tan are just side ratios. Everything else is bookkeeping.
sin θ = opp/hypcos θ = adj/hyptan θ = opp/adjFor non-right triangles, two tools finish the job: the law of sines (a/sin A = b/sin B) and the law of cosines (c² = a² + b² − 2ab cos C). The identity sin²θ + cos²θ = 1 is the Pythagorean theorem in disguise.
The skills, taught in order
3.1 Angles: degrees and radians
Radians are not optional. They are the unit that makes arc length, angular velocity, and small-angle work behave. Convert with π rad = 180°.
θrad = θdeg × π/180s = rθSo 30° = 30 × π/180 = π/6 = 0.524 rad. On a wheel of radius 2 m, turning through 0.524 rad moves a rim point s = 2 × 0.524 = 1.05 m. Feed degrees into s = rθ and the answer comes out about 57 times too large.
3.2 The three ratios and the special angles
SOH-CAH-TOA defines them; the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) recover the angle from a ratio. Memorise these five angles and you can sketch any wave by hand.
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undef. |
3.3 Solving any triangle
A right triangle is fixed by any two independent pieces of data. Every other triangle needs one of two laws. Choose by what you are given:
| You are given | Use |
|---|---|
| A right angle is present | SOH-CAH-TOA |
| A side with its opposite angle, plus one more piece | Law of sines: a/sin A = b/sin B |
| Two sides and the included angle, or all three sides | Law of cosines: c² = a² + b² − 2ab cos C |
The law of cosines collapses into Pythagoras when C = 90°, because cos 90° = 0.
3.4 The identities worth knowing
You need very few. The Pythagorean identity and the two double-angle forms cover almost all engineering use:
sin²θ + cos²θ = 1sin 2θ = 2 sin θ cos θcos 2θ = cos²θ − sin²θThe first is just Pythagoras on a hypotenuse of length 1. It is what lets you recover cos θ from a known sin θ without a second measurement.
3.5 Periodic motion: the language of vibration
Every oscillation, a vibrating panel, an AC voltage, a travelling wave, is written in one standard form:
x(t) = A sin(ωt + φ)A is the amplitude (peak value), ω is the angular frequency in rad/s, and φ is the phase, fixing where the cycle starts. These connect to ordinary frequency and period by ω = 2πf and T = 1/f. A 10 Hz vibration has ω = 2π(10) = 62.8 rad/s and repeats every T = 0.1 s.
Engineering connection: Statics, Dynamics, Vibrations, Machine Design. Statics Module 2 (Force Vectors) assumes this chapter cold.
Worked example: the loading ramp
A loading ramp rises 1.5 m over a horizontal run of 4.0 m. Find the ramp angle, the ramp length, and the component of a 600 N crate weight acting along the ramp surface.
- ProblemFind θ, L, and the along-ramp weight component for the ramp in Figure 1.
- Given / findRise 1.5 m, run 4.0 m, W = 600 N. Find θ, L, W sin θ.
- AssumptionsStraight, rigid ramp; weight acts straight down.
- ModelOne right triangle: opposite = 1.5 m, adjacent = 4.0 m. The weight splits along and normal to the slope using the same angle θ.
- Equationsθ = tan⁻¹(opp/adj) L = √(run² + rise²) Falong = W sin θ
- Solveθ = tan⁻¹(1.5/4.0) = 20.6°. L = √(16 + 2.25) = √18.25 = 4.27 m. Along the ramp: 600 sin 20.6° = 211 N; normal to it: 600 cos 20.6° = 562 N.
- Checksin 20.6° = 1.5/4.27 = 0.351 directly from the sides. Components: √(211² + 562²) = 600 N, the original weight. A 20° ramp feeling about one third of the weight as pull is consistent with experience.
- ConclusionWhoever holds a crate on this ramp resists 211 N, not 600 N. This split of weight into along-slope and normal components is performed in nearly every Statics and Dynamics problem with an incline.
Worked example 2: read a vibration signal
A sensor records a machine vibration as x(t) = 3 sin(20πt + π/4) mm, with t in seconds. Find the amplitude, angular frequency, ordinary frequency, period, phase angle, and the displacement at t = 0.
- Given / findx(t) = 3 sin(20πt + π/4) mm. Find A, ω, f, T, φ, and x(0).
- Match the standard formCompare with x = A sin(ωt + φ): A = 3 mm, ω = 20π rad/s, φ = π/4.
- Frequency and periodf = ω/2π = 20π/2π = 10 Hz, so T = 1/f = 0.10 s.
- Phase in degreesφ = π/4 = 45°, so the signal is already one eighth of a cycle along at t = 0.
- Displacement at t = 0x(0) = 3 sin(π/4) = 3 × 0.707 = 2.12 mm.
- CheckThe amplitude is 3 mm, so x(0) must lie between −3 and +3 mm; 2.12 mm does. A nonzero start value is exactly what a phase shift produces.
- ConclusionReading A, ω, and φ straight off a signal is the daily skill in vibrations and signal work. The same three numbers describe a shaking foundation, an AC supply, or a sound wave.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Calculator in the wrong angle mode | sin 30 = −0.988 instead of 0.5 | "Does sin 30° return exactly 0.5 right now?" | Run that one-key test at the start of every session and exam. |
| SOH-CAH-TOA on a non-right triangle | Sides that do not close; impossible answers | "Where is the 90° angle in my triangle?" | No right angle means law of sines or law of cosines. Always. |
| Opposite and adjacent swapped | Components reversed; angle complement appears | "Opposite and adjacent relative to which angle?" | Mark the angle first, then label sides relative to it. Redraw if unsure. |
| Degrees fed into s = rθ | Arc lengths 57 times too large | "Is my angle in radians for this formula?" | Arc length, angular velocity, and small-angle work demand radians. Convert first. |
Practice ladder
A right triangle has opposite 5 and adjacent 12. Find the hypotenuse and all three ratios of θ.
Show answer
Hypotenuse = 13 (the 5-12-13 triple). sin θ = 5/13 = 0.385, cos θ = 12/13 = 0.923, tan θ = 5/12 = 0.417; θ = 22.6°.
Then convert 135° to radians and find the arc length it sweeps on a radius of 0.40 m.
Show answer
135° = 135 × π/180 = 2.356 rad. Arc length s = rθ = 0.40 × 2.356 = 0.942 m. The angle must be in radians for s = rθ.
Two rods of lengths 5 m and 7 m meet at a 60° joint. Find the distance between their far ends (law of cosines).
Show answer
c² = 5² + 7² − 2(5)(7) cos 60° = 25 + 49 − 35 = 39, so c = 6.24 m. The cos 60° = 0.5 makes this one clean.
A vibration is x(t) = 5 sin(4πt) mm. State its amplitude, frequency, and period.
Show answer
A = 5 mm; ω = 4π rad/s so f = ω/2π = 2 Hz; and T = 1/f = 0.5 s. With no phase term it starts at zero.
From a point on flat ground the top of a mast has an elevation angle of 32°. Walking 20 m closer, the angle becomes 40°. Find the mast height.
Show answer
h/tan 32° − h/tan 40° = 20, so h(1.6003 − 1.1918) = 20, giving h = 20/0.4086 = 49.0 m. Two triangles sharing one unknown: the standard surveying pattern.
Measure a real ramp, staircase, or roof with a phone inclinometer and a tape measure. Compute angle from rise and run, compare with the phone reading, then compute what fraction of a known weight acts along the slope.
What good work looks like
Both angle values with their deviation explained (measurement noise, where the phone sat), the W sin θ computation, and a sentence connecting it to ramp safety or stair comfort.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Create a one-page "Triangle Toolkit": the right-triangle ratio diagram, law of sines and cosines each with one worked mini-example you wrote yourself, the special-angle table (30°, 45°, 60°), and the A sin(ωt + φ) anatomy labeled (amplitude, frequency, phase).
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. Convert 180° and 90° to radians, and 1 rad to degrees.
180° = π rad; 90° = π/2 rad; 1 rad = 180/π ≈ 57.3°.
2. When do you need the law of cosines rather than the law of sines?
When you know two sides and the included angle, or all three sides. The law of sines needs a known side-angle opposite pair.
3. State the Pythagorean identity and its origin.
sin²θ + cos²θ = 1: the Pythagorean theorem applied to a hypotenuse of length 1.
4. In A sin(ωt + φ), name each symbol's physical meaning.
A: amplitude (peak value). ω: angular frequency in rad/s. φ: phase shift, where the cycle starts. This is the standard form of every vibration signal.
5. A 600 N weight sits on a θ incline. What are the components along and normal to the surface?
Along: W sin θ. Normal: W cos θ. At 20.6°: 211 N and 562 N.
Textbook mapping
| Item | Mapping |
|---|---|
| Main sources | Stewart, Redlin and Watson, Precalculus: Mathematics for Calculus (trigonometry chapters); Stewart, Calculus: Early Transcendentals (review appendix) |
| Core topics | 2.1 Angles and radians · 2.2 sin, cos, tan · 2.3 Right triangles · 2.4 Laws of sines and cosines · 2.5 Identities · 2.6 Phase, amplitude, periodic motion · 2.7 Geometry of beams, slopes, arcs, circular motion |
| Engineering connection | Statics, Dynamics, Vibrations, Machine Design. Statics Module 2 (Force Vectors) assumes this chapter cold. |
| Skip on first pass | Identity manipulation beyond the Pythagorean and double-angle pair; spherical geometry. |
| Read next | Vectors and Coordinate Systems. |