Trigonometry for Physics — SOHCAHTOA, Components and Angles

High School⏱ 9 min readMath for Physics

Trigonometry for physics: SOHCAHTOA, resolving vectors into components, finding angles, and the degrees-vs-radians trap.

Quick answer

Trigonometry relates a right triangle's angles to its side ratios: sine, cosine and tangent. In physics its main job is splitting vectors into x and y components — and rebuilding them afterward.

$\sin\theta = \frac{\text{opp}}{\text{hyp}},\quad \cos\theta = \frac{\text{adj}}{\text{hyp}},\quad \tan\theta = \frac{\text{opp}}{\text{adj}}$

Unit: ratios are unitless; angles in degrees or radians

Example: A 50 N force at 30° above horizontal has Fx = 50cos30° ≈ 43.3 N and Fy = 50sin30° = 25 N.

What you will learn

Use SOHCAHTOA to find unknown sides of right triangles
Resolve any vector into x and y components
Find angles with inverse trig functions
Avoid the degrees-vs-radians calculator trap

SOHCAHTOA: the three ratios

Every physics student meets the same triangle: a vector as the hypotenuse, its x and y components as the legs. Trigonometry is the dictionary translating between them.

In a right triangle with acute angle $\theta$ :

Sine: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$ — the SOH
Cosine: $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ — the CAH
Tangent: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$ — the TOA

'SOHCAHTOA' is just those three sentences compressed. Given any two pieces of a right triangle (a side and an angle, or two sides), trig finds the rest.

\sin\theta = \frac{\text{opp}}{\text{hyp}},\quad \cos\theta = \frac{\text{adj}}{\text{hyp}},\quad \tan\theta = \frac{\text{opp}}{\text{adj}

opp/adj are relative to the angle theta you are using

Resolving vectors: trig's main job in physics

The killer application: resolving a vector into components. If vector $\vec{A}$ has magnitude $A$ at angle $\theta$ above the +x axis:

$A_x = A\cos\theta, \qquad A_y = A\sin\theta$

Why does physics love this? Because components are plain signed numbers. Forces, velocities, fields — once resolved, they add and subtract like ordinary arithmetic, per axis independently. The vector nature is fully preserved: nothing is lost in the split, and $A = \sqrt{A_x^2 + A_y^2}$ rebuilds the original exactly.

To go the other way — components to angle — use the inverse: $\theta = \tan^{-1}(A_y / A_x)$ , then fix the quadrant by sketching.

Worked mini-example: pulling a sled

A child pulls a sled with a 50 N force at 30° above the horizontal. Find the horizontal and vertical components.

Identify: hypotenuse = 50 N, angle from horizontal = 30°
Horizontal (adjacent): $F_x = 50\cos30° = 50 \times 0.866 \approx 43.3\ \mathrm{N}$
Vertical (opposite): $F_y = 50\sin30° = 50 \times 0.5 = 25.0\ \mathrm{N}$
Check: $\sqrt{43.3^2 + 25^2} \approx 50\ \mathrm{N}$ ✓

Physics reading: only 43.3 N actually pulls the sled forward; the 25 N upward component lightens the sled's contact with the snow, reducing friction. That is why pulling at an angle beats pulling horizontally — trig quantifies the bonus.

Key idea

Components separate a vector's jobs. On a ramp, the parallel component causes sliding while the perpendicular component presses into the surface. Trig tells you exactly how much does which.

The two traps: modes and mislabeled sides

Two traps catch everyone eventually:

Degree/radian mode. Your calculator has a D/R indicator — check it every time. $\sin(30°) = 0.5$ but $\sin(30\ \text{radians}) = -0.988$ . In physics classes, degrees dominate until calculus-heavy work, where radians take over.
Which side is 'opposite'? Opposite and adjacent are defined relative to your angle. Redraw and relabel for each new angle instead of memorizing 'sine = vertical' — that shortcut breaks the moment the angle is measured from the vertical.

Variables and units

Symbol	Name	Unit	Meaning
$sin θ$	Sine	unitless	Ratio opposite/hypotenuse
$cos θ$	Cosine	unitless	Ratio adjacent/hypotenuse
$tan θ$	Tangent	unitless	Ratio opposite/adjacent
$θ$	Angle	degrees or radians	Angle, usually from the +x axis

Common mistakes

✗ Trigonometry only works for right triangles, so it is useless for general physics.

Component resolution always builds right triangles — that is the point. And the laws of sines and cosines extend trig to any triangle. Right-triangle trig covers the vast majority of physics problems.

✗ My calculator gave a weird angle, so the physics must be wrong.

Almost certainly the calculator is in radian mode while you think in degrees (or vice versa). A 30° angle entered as 30 radians is nonsense. Check the mode indicator before doubting the physics.

✗ Sine is for vertical and cosine is for horizontal, always.

Only when the angle is measured from the horizontal. Measure from the vertical and they swap. The safe habit: sketch the triangle, label opposite/adjacent relative to your angle, then choose.

Real-world applications

Surveyors resolve slopes into horizontal distances and vertical rises with tangent.
Engineers split forces into components to analyze structures and ramps.
Navigators convert bearings and distances into north/east components.
Optics uses Snell's law, built on sines, to trace light through lenses.

Frequently asked questions

What is SOHCAHTOA?

SOHCAHTOA is a mnemonic for the three basic ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Pick the ratio containing the side you know and the side you want.

How do you find an angle from two sides?

Use the inverse functions: theta = sin^-1(opposite/hypotenuse), and similarly for cos^-1 and tan^-1. Make sure your calculator is in the right angle mode first.

What does 'resolving a vector' mean?

Resolving a vector into components means finding its x and y parts: Ax = A cos(theta), Ay = A sin(theta), with theta from the x-axis. Components turn vector problems into ordinary scalar arithmetic.

Which angle do I use in the formulas?

The angle must be measured from the correct reference. In physics, theta is usually from the +x axis; in navigation, bearings are from north. Convert first, then apply SOHCAHTOA.