Workshop Calculation: Trigonometry and Its Workshop Applications
Trigonometry in the workshop
This domain tests right-triangle ratios, Pythagoras’ theorem and inverse trigonometric functions applied to tapers, sine bars, inclines and setting-out dimensions. The exam expects correct ratio selection and degree-mode calculation.
Core rules
- SOHCAHTOA: sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.
- Pythagoras: hypotenuse² = opposite² + adjacent².
- Use sin⁻¹, cos⁻¹ or tan⁻¹ to find an angle from a ratio.
- For a taper, the half-angle α satisfies tan α = (D − d)/(2L), where D and d are the large and small diameters and L is the taper length. The included angle is 2α.
- Sine-bar setting: h = L sin θ, where L is the roller centre distance and h is the slip-gauge height.
Worked example
A sine bar has roller centre distance L = 200 mm. It must be set to θ = 15°. Slip-gauge height h = L sin θ = 200 × 0.2588 = 51.76 mm. Reasoning: the sine bar, the surface plate and the gauge stack form a right triangle; the gauge height is opposite the set angle and the roller centre distance is the hypotenuse, so sine is the correct ratio.
Traps
- Leaving the calculator in radian mode instead of degree mode.
- Using the included taper angle when the formula needs the half-angle.
- Mixing up opposite and adjacent when choosing tan instead of sin or cos.
- Using the slant length where the vertical height is required.
- Rounding the trigonometric ratio too early and losing accuracy.