Trigonometry & Radians — Maths for Organic Chemistry

Angles, triangles and waves

The language of geometry and everything periodic

Trigonometry links angles to ratios of sides. SOHCAHTOA captures the three core functions, and the unit circle extends them to any angle — including the obtuse and reflex angles that turn up in molecular geometry and diffraction.

sin θ = \frac{opp}{hyp}, cos θ = \frac{adj}{hyp}, tan θ = \frac{opp}{adj}

Chemistry measures angles in radians because that is what calculus expects: the clean identity $d (sin θ) / d θ = cos θ$ holds only in radians. The conversion is $1 rad = 18 0^{\circ} / π \approx 57. 3^{\circ}$ . Sine and cosine then describe every wave — light, sound, quantum wavefunctions — with just three parameters: amplitude, frequency and phase.

The most important trig formula in analytical chemistry is Bragg's law, $nλ = 2 d sin θ$ , which gives the spacing between crystal planes from the angle at which X-rays constructively interfere. Knowing $sin θ$ at the diffraction peak is enough to deduce an atomic spacing to picometre precision.

Core identities and conversions

Degree ↔ radian

θ_{rad} = \frac{π}{180} θ_{deg}

Pythagorean identity

sin^{2} θ + cos^{2} θ = 1

Double-angle

sin 2 θ = 2 sin θ cos θ

Bragg's law

nλ = 2 d sin θ

General wave

y = A sin (k x + ϕ)

Key values

sin 3 0^{\circ} = \frac{1}{2}, sin 4 5^{\circ} = \frac{2}{2}, sin 6 0^{\circ} = \frac{3}{2}

Crystallography and wave extras

Cubic interplanar spacing

d_{hk l} = \frac{a}{h ^{2} + k ^{2} + l ^{2}}

where (hkl) are Miller indices and

a

is the cell edge.

Higher-order Bragg reflections

nλ = 2 d sin θ

. The 2nd-order reflection at spacing

d

is equivalent to a 1st-order reflection from planes at

d /2

Small-angle approximation

For

θ ≪ 1 rad

sin θ \approx θ

cos θ \approx 1

tan θ \approx θ

. Used in Bragg for neutron small-angle scattering (SANS) and NMR lineshape analysis.

Wave superposition

A sin (k x + ϕ_{1}) + A sin (k x + ϕ_{2}) = 2 A cos (\frac{Δ ϕ}{2}) sin (k x + \frac{ϕ _{1} + ϕ _{2}}{2})

Common pitfalls

Calculator in degrees when radians are needed (or vice versa). Bragg's law uses $sin θ$ , which is dimensionless and the same in both systems — but when evaluating integrals or derivatives involving trig, the variable must be in radians. Check your calculator mode before every calculation.
Using cos instead of sin in Bragg's law. The correct form is $nλ = 2 d sin θ$ where $θ$ is the glancing angle (angle between the beam and the plane surface). A common mistake is to use the complement or write $cos θ$ , which is valid only when $θ$ is measured from the normal rather than the surface.
Sign/convention errors with phase. A phase shift of $+ ϕ$ shifts the wave left (earlier in time/space), while $- ϕ$ shifts it right. Mixing sign conventions when adding two waves can give a phase difference of $2 ϕ$ instead of zero, incorrectly predicting constructive vs destructive interference.
Forgetting the factor of 2 in the phase-difference formula. A path difference of $Δ L$ corresponds to a phase difference of $Δ ϕ = 2 π Δ L / λ$ , not $π Δ L / λ$ . Missing the factor of $2 π$ halves the predicted phase, turning what should be destructive interference into constructive.

Where it shows up

Bond angles, Bragg's law

nλ = 2 d sin θ

for X-ray crystallography, the wave nature of electrons, and the phase of orbitals in bonding. Open the unit circle and wave explorer to see these in action.

Going deeper

The Miller-index spacing formula $d_{hk l} = a / h^{2} + k^{2} + l^{2}$ combines with Bragg's law to give a direct route from peak positions to the lattice parameter. Each observed reflection has a unique combination of integers (hkl); plotting $sin^{2} θ$ against $h^{2} + k^{2} + l^{2}$ for a cubic crystal gives a straight line of slope $λ^{2} / (4 a^{2})$ . The systematic absences — certain (hkl) combinations that give no reflection in bcc or fcc — are a direct consequence of destructive interference from atoms at the cell centre or face centres, and they are what allows crystallographers to distinguish the cell type from a powder diffraction pattern.

Unit circle

Sweep the angle and read sine, cosine and tangent directly off the circle. sin²θ + cos²θ = 1 holds at every point.

Sweep the angle and read sine, cosine and tangent off the circle

angle θ35° = 0.61 rad

sin θ0.574

cos θ0.819

tan θ0.700

sin^{2} θ + cos^{2} θ = 1

The green segment is cos θ, the blue is sin θ — coordinates of a point walking the circle. They always satisfy sin²+cos²=1, the Pythagorean identity.

Worked example 1Degrees to radians and back — bond angles

A tetrahedral molecule has a bond angle of 109.47°. Express this in radians. Also convert the octahedral angle of 90° to radians.

Worked example 2Bragg's law — finding a d-spacing

X-rays of wavelength $λ = 154.0 pm$ (Cu Kα) diffract from a set of NaCl planes at a first-order ( $n = 1$ ) Bragg angle of $θ = 15.8 7^{\circ}$ . Find the plane spacing $d$ .

Wave explorer

Adjust amplitude, frequency and phase. Two waves with a phase difference of π cancel completely — the maths of destructive interference.

Amplitude, frequency and phase of a sine wave

y = 1.0 sin (1.0 x + 0.0)

amplitude A1.0

frequency k1.0

phase φ0.0 rad

Amplitude sets the height, frequency sets how many waves fit, and phase slides it left or right. Add two waves with a phase difference of π and they cancel — the maths of interference and bonding.

Worked example 3Amplitude and phase in an NMR free-induction decay

An NMR free-induction decay (FID) is modelled as $y (t) = 0.80 sin (2 π \times 200 t + π /4)$ where $t$ is in seconds. Identify the amplitude, frequency (in Hz), and phase shift (in degrees).

Worked example 4Constructive vs destructive interference of electron waves

Two atomic orbitals are described by $ψ_{1} = sin (k x)$ and $ψ_{2} = sin (k x + ϕ)$ . Their sum is the bonding combination and their difference the antibonding. What value of $ϕ$ gives maximum constructive interference, and what value gives complete cancellation?

Worked examples

Pull Bragg's law, SOHCAHTOA and wave interference together — try each before revealing the solution.

Worked example 5Bond length from a bond angle via SOHCAHTOA

In a tetrahedral CH₄ molecule the C–H bond length is 109 pm and the bond angle is 109.47°. Using the bisector of the H–C–H angle, find the perpendicular distance from C to the line joining two H atoms.

Worked example 6Finding the order of diffraction from a known d-spacing

Silicon (111) planes have spacing $d = 313.6 pm$ . At what angle $θ$ does second-order ( $n = 2$ ) Bragg diffraction occur for Cu Kα radiation ( $λ = 154.2 pm$ )?

Worked example 7Phase difference in a two-slit UV absorption spectrum

Two coherent light beams travel paths of $L_{1} = 500 nm$ and $L_{2} = 625 nm$ before recombining. The wavelength is $λ = 500 nm$ . Calculate the phase difference and state whether the interference is constructive or destructive.

Worked example 8Cubic d-spacing from Miller indices

Platinum (fcc, $a = 392.4 pm$ ) gives a Bragg reflection at $θ = 23. 4^{\circ}$ with Cu Kα radiation ( $λ = 154.0 pm$ ). Find the d-spacing and hence the Miller indices (hkl) of the reflecting planes.

Worked example 9Small-angle approximation in SAXS

In small-angle X-ray scattering (SAXS) a polymer forms a lamellar repeat with spacing $d = 12.0 nm$ . The first-order Bragg peak occurs at scattering angle $2 θ = 0.73 6^{\circ}$ for $λ = 0.154 nm$ . Verify Bragg's law exactly, then show the small-angle approximation ( $sin θ \approx θ$ in radians) gives the same answer to three significant figures.

ChallengeIndex two diffraction peaks to determine the lattice parameter

A powder diffraction pattern of a cubic metal recorded with Mo Kα radiation ( $λ = 71.07 pm$ ) shows first-order peaks at $θ_{1} = 10.0 8^{\circ}$ and $θ_{2} = 14.3 4^{\circ}$ . Index these two peaks (assign Miller indices), deduce the lattice parameter $a$ , and identify the cell type (sc, bcc, or fcc) from the systematic absences.

Check yourself

Four quick questions on angles, waves and Bragg's law.

Question 1 of 4 · Score 0

How many radians are in 180°?

Choose an answer.