Physics › Maths bridge › Trigonometry and resolving vectors
Trigonometry and resolving vectors
Sine, cosine and tangent at working speed, and splitting a vector into components.
Resolving a vector into components
For a vector of magnitude F at angle θ to an axis, the component parallel to that axis is F cos θ and the component perpendicular to it is F sin θ. Adjacent takes the cosine. The same construction covers every resolving problem in mechanics, fields and waves.
Use the calculator mode that matches the angle's unit. Angles stated in degrees require degree mode; phase arguments such as ωt are measured in radians. The wrong mode returns a plausible number with no warning, so check the mode every time trigonometry appears.
The components in practice
WORKED EXAMPLE
Resolving a pull
A rope pulls a sledge with a force of 20 N at 35° above the horizontal. Find the horizontal and vertical components.
The angle is measured from the horizontal, so the horizontal component takes the cosine. 20 cos 35° = 16.4 N.
The vertical component takes the sine: 20 sin 35° = 11.5 N. As a check, 16.42 + 11.52 ≈ 202, so the two components carry the full magnitude of the vector.
GUIDED PRACTICE
Weight on a slope
A 2.0 kg block rests on a 30° incline. Resolve its weight into the component along the slope and the component pressing into it, deciding first which one takes the sine.
Show the working
The weight is mg = 2.0 × 9.81 = 19.6 N, and the angle between the weight and the perpendicular to the slope equals the slope angle.
Along the slope, mg sin 30° = 9.8 N. Into the slope, mg cos 30° = 17.0 N.
Where physics leans on this: Scalars and vectors · Projectile motion · Flux and flux linkage. All eight skills: the maths bridge.