Maths bridge › Trigonometry and resolving vectors
Trigonometry and resolving vectors
Sine, cosine and tangent at working speed, and splitting a vector into components.
The one diagram that matters
A vector at angle θ to a reference direction splits into two components: cos θ along the direction the angle is measured from, and sin θ perpendicular to it. Adjacent takes the cosine. Every resolving problem in mechanics, fields and waves is this one diagram wearing different clothes.
Check the calculator's angle mode every time trigonometry appears: degrees for geometry, radians when ω and phase turn up in periodic motion. A right answer in the wrong mode is the quietest error in physics.
Worked and handed over
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 rebuilds 202: the components still contain the whole vector.
YOUR TURN
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, before opening the working.
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. On an incline the sine goes with the slide.
Where physics leans on this: Scalars and vectors · Projectile motion · Flux and flux linkage. All eight skills: the maths bridge.