Measuring angles: degree and radian
An angle can be quoted in two units. The degree divides a full turn into 360 equal parts and is what protractors and everyday speech use. The radian is defined geometrically: the angle subtended at the centre of a circle by an arc whose length equals the radius. Because the full circumference is \(2\pi r\), one full turn is \(2\pi\) radians, which fixes the bridge between the two systems: \(360^\circ = 2\pi\) rad, or equivalently \(180^\circ = \pi\) rad.
The radian is not a cosmetic preference. It is the only unit in which the calculus of trig functions is clean — \(\dfrac{d}{dx}(\sin x)=\cos x\) holds only when \(x\) is in radians — and it is the only unit in which the small-angle approximations are true. Physics therefore defaults to radians whenever a derivative, an integral, or an approximation is in play. Degrees survive mainly when reading a value off a diagram or quoting a standard angle.
Degree ↔ radian conversion
Degrees to radians: multiply by \(\pi/180\). Radians to degrees: multiply by \(180/\pi\). Worth memorising: \(30^\circ=\pi/6\), \(45^\circ=\pi/4\), \(60^\circ=\pi/3\), \(90^\circ=\pi/2\), and 1 rad ≈ 57.3°.
Two derived facts repay attention. The arc length intercepted by an angle \(\theta\) (in radians) on a circle of radius \(r\) is simply \(s = r\theta\); this is the workhorse of rotational kinematics, where it becomes \(v = r\omega\). And an angle small in radians is a small pure number — \(6^\circ\) is only \(0.105\) rad — which is exactly why the approximations below behave so well.
The six trigonometric ratios
For a right-angled triangle with an acute angle \(\theta\), label the side opposite \(\theta\) as opposite, the side next to it (not the hypotenuse) as adjacent, and the longest side as hypotenuse. The three primary ratios are sine, cosine and tangent; the other three — cosecant, secant and cotangent — are their reciprocals. In physics you will use sine, cosine and tangent almost exclusively, but NEET expects you to recognise all six.
Primary ratios
\(\sin\theta\) = opposite / hypotenuse
\(\cos\theta\) = adjacent / hypotenuse
\(\tan\theta\) = opposite / adjacent
Reciprocal ratios
\(\csc\theta\) = 1 / sin θ
\(\sec\theta\) = 1 / cos θ
\(\cot\theta\) = 1 / tan θ = cos θ / sin θ
The single most useful physics application of these ratios is the resolution of a vector. A vector of magnitude \(A\) making angle \(\theta\) with the \(+x\) axis has components \(A_x = A\cos\theta\) and \(A_y = A\sin\theta\); the cosine takes the angle measured from the axis you are projecting onto. This is the bedrock of projectile motion, inclined planes and force balance, and it is why the standard-angle values below have to be instant.
Standard-angle value table
Five angles — \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) — recur so often that their sine, cosine and tangent must be recalled without hesitation. A clean memory aid: write sine as \(\sqrt{n}/2\) for \(n = 0,1,2,3,4\), giving \(0,\ \tfrac12,\ \tfrac{1}{\sqrt2},\ \tfrac{\sqrt3}{2},\ 1\). Cosine is the same list read backwards, and tangent is sine divided by cosine.
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | ∞ (undefined) |
Note the boundaries. At \(0^\circ\) the opposite side vanishes, so sine is zero and cosine is one; at \(90^\circ\) the roles swap, and tangent — being \(\sin/\cos\) — diverges because cosine is zero. These limits are not trivia: a horizontally launched projectile (\(\theta = 0\)) has no vertical velocity component because \(\sin 0 = 0\), and a vertically thrown ball (\(\theta = 90^\circ\)) has no horizontal component because \(\cos 90^\circ = 0\).
Signs in the four quadrants
Beyond \(90^\circ\), the ratios take signs that depend on the quadrant. Picture the angle measured anticlockwise from the \(+x\) axis on a unit circle; the point on the circle has coordinates \((\cos\theta,\ \sin\theta)\). Cosine therefore carries the sign of the \(x\)-coordinate and sine the sign of the \(y\)-coordinate, with tangent following as their ratio. The standard mnemonic is ASTC — "All Students Take Coffee" — naming which ratios are positive in quadrants I, II, III, IV in turn.
Key identities
A handful of identities recur across mechanics, optics, oscillations and AC. They are not to be re-derived in the exam hall; they are to be recalled. The foundation is the Pythagorean identity, a restatement of \(a^2+b^2=c^2\) on the unit circle: \(\sin^2\theta+\cos^2\theta=1\). From it follow the compound-angle (addition) formulas and the double-angle formulas, which let you collapse \(\sin^2\theta\) and \(\cos^2\theta\) into linear cosines — the trick behind integrating \(\sin^2 x\) and computing the RMS value of an alternating current.
| Identity | Form | Where it surfaces |
|---|---|---|
| Pythagorean | sin²θ + cos²θ = 1 | Recovering a missing component; magnitude from components |
| Sine sum | sin(A±B) = sinA cosB ± cosA sinB | Superposition of waves; interference |
| Cosine sum | cos(A±B) = cosA cosB ∓ sinA sinB | Phase combination; beat phenomena |
| Double angle (sin) | sin 2θ = 2 sinθ cosθ | Maximum range of a projectile (sin 2θ = 1) |
| Double angle (cos) | cos 2θ = 1 − 2sin²θ = 2cos²θ − 1 | Power reduction → ∫sin²x, AC RMS |
The projectile range formula is a clean payoff of the double-angle identity. The horizontal range of a projectile launched at speed \(u\) and angle \(\theta\) is \(R = \dfrac{u^2\sin 2\theta}{g}\). Because \(\sin 2\theta\) peaks at \(1\) when \(2\theta = 90^\circ\), the range is maximum at \(\theta = 45^\circ\) — and complementary angles such as \(30^\circ\) and \(60^\circ\) give equal ranges because \(\sin 60^\circ = \sin 120^\circ\). One identity explains two standard NEET facts.
These identities power wave superposition — see Oscillations for how \(\sin(\omega t+\phi)\) combines into SHM and beats.
Small-angle approximations
When an angle is small and measured in radians, the trig functions simplify dramatically. The functions have power-series expansions — \(\sin\theta = \theta - \tfrac{\theta^3}{6}+\dots\), \(\cos\theta = 1 - \tfrac{\theta^2}{2}+\dots\), \(\tan\theta = \theta + \tfrac{\theta^3}{3}+\dots\) — and for small \(\theta\) the higher powers are negligible. Keeping only the leading terms gives the three approximations every physics chapter leans on. They are accurate to better than one percent for \(\theta\) up to about \(0.1\) rad (roughly \(6^\circ\)), and this is no mere convenience: without them, the simple pendulum, paraxial optics and thin-prism deviation would have no closed-form NEET formulas at all.
Small-angle approximations (θ in radians, θ ≪ 1)
For \(\theta \lesssim 0.1\) rad (about \(6^\circ\)): \(\sin\theta\approx\theta\), \(\tan\theta\approx\theta\), and \(\cos\theta\approx 1-\tfrac{\theta^2}{2}\) (often simplified to \(\cos\theta\approx 1\)). Use the \(\theta^2/2\) form whenever the leading "1" cancels and the small correction is what matters.
cos θ ≈ 1
drop the correction
when cos θ appears alone, not as a difference
- Component nearly equal to the full magnitude
- Normal force on a gentle incline
- Quick magnitude estimates
cos θ ≈ 1 − θ²/2
keep the correction
when the "1" cancels and a small term remains
- Height of a pendulum bob: \(h=\ell(1-\cos\theta)\approx \tfrac12\ell\theta^2\)
- Potential energy of a small swing
- Path-difference corrections in optics
Why this matters can be seen in one line. The exact equation of motion of a simple pendulum is \(m\ell\ddot\theta = -mg\sin\theta\), which is not the equation of simple harmonic motion because of the \(\sin\theta\). Replace \(\sin\theta\) by \(\theta\) (valid for small swings) and it becomes \(\ddot\theta = -\tfrac{g}{\ell}\theta\) — the canonical SHM equation, with angular frequency \(\omega=\sqrt{g/\ell}\) and period \(T=2\pi\sqrt{\ell/g}\). The familiar pendulum formula is, at heart, a small-angle approximation, and it quietly stops being accurate for large amplitudes.
Worked examples
A force of magnitude \(20~\text{N}\) acts at \(60^\circ\) above the horizontal. Resolve it into horizontal and vertical components.
Set up. The horizontal component is \(F_x = F\cos\theta\) and the vertical component is \(F_y = F\sin\theta\), with \(\theta = 60^\circ\) measured from the horizontal axis.
Apply the standard values. \(\cos 60^\circ = \tfrac12\) and \(\sin 60^\circ = \tfrac{\sqrt3}{2}\).
Compute. \(F_x = 20\times\tfrac12 = 10~\text{N}\); \(F_y = 20\times\tfrac{\sqrt3}{2} = 10\sqrt3 \approx 17.3~\text{N}\). A quick check: \(\sqrt{F_x^2+F_y^2}=\sqrt{100+300}=20~\text{N}\), recovering the original magnitude via the Pythagorean identity.
A simple pendulum of length \(\ell = 1~\text{m}\) is displaced by \(5^\circ\) from the vertical. Using the small-angle approximation, estimate (a) the horizontal displacement of the bob and (b) its rise in height. Take \(g = 9.8~\text{m s}^{-2}\).
Convert the angle. \(\theta = 5^\circ \times \dfrac{\pi}{180} \approx 0.0873~\text{rad}\) — well below \(0.1\) rad, so the approximations apply.
(a) Horizontal displacement. \(x = \ell\sin\theta \approx \ell\theta = 1\times 0.0873 = 0.0873~\text{m} \approx 8.7~\text{cm}\).
(b) Rise in height. Here the leading "1" cancels, so keep the quadratic cosine: \(h = \ell(1-\cos\theta)\approx \ell\cdot\tfrac{\theta^2}{2} = 1\times\dfrac{(0.0873)^2}{2}\approx 3.8\times10^{-3}~\text{m} \approx 3.8~\text{mm}\).
Note. The period is independent of this small amplitude: \(T = 2\pi\sqrt{\ell/g}=2\pi\sqrt{1/9.8}\approx 2.0~\text{s}\). The example shows both small-angle forms in one problem — \(\sin\theta\approx\theta\) for displacement, \(\cos\theta\approx1-\theta^2/2\) for height.
A car of mass \(m\) takes a curve of radius \(r\) on a road banked at a small angle \(\theta\). Show that the design speed for which no friction is needed satisfies \(v^2 \approx rg\theta\) for small \(\theta\).
Exact result. For a frictionless banked curve, balancing the horizontal component of the normal force against the centripetal requirement gives \(\tan\theta = \dfrac{v^2}{rg}\), so \(v^2 = rg\tan\theta\).
Apply the small-angle approximation. For a gently banked road \(\theta\) is small (in radians), so \(\tan\theta\approx\theta\).
Result. \(v^2 \approx rg\theta\). For example, \(r = 100~\text{m}\), \(\theta = 0.1~\text{rad}\) (about \(5.7^\circ\)), \(g = 10~\text{m s}^{-2}\) give \(v^2 \approx 100\times10\times0.1 = 100\), i.e. \(v \approx 10~\text{m s}^{-1}\). The approximation turns a tangent into a clean linear relation between design speed and banking angle.
Where this shows up in NEET physics
Trigonometry across the NEET syllabus
Mathematical Tools is not graded directly, but every tool below is assumed by chapters that are tested. These are the application points NEET keeps reusing.
- Motion in a Plane — projectile velocity split as \(u\cos\theta\) and \(u\sin\theta\); range \(R=u^2\sin 2\theta/g\) maximal at \(45^\circ\).
- Laws of Motion — inclined planes resolve weight into \(mg\sin\theta\) along and \(mg\cos\theta\) perpendicular to the slope.
- Vectors & equilibrium — forces beyond \(90^\circ\) carry negative components; ASTC fixes the sign.
- Alternating Current — phase angles span all four quadrants in phasor diagrams.
- Oscillations — the simple pendulum becomes SHM only after \(\sin\theta\to\theta\), giving \(T=2\pi\sqrt{\ell/g}\).
- Wave Optics — Young's double-slit fringe spacing uses \(\sin\theta\approx\tan\theta\approx y/D\).
- Laws of Motion — banking of roads: \(v^2=rg\tan\theta\approx rg\theta\) for gentle banks.
- Ray Optics — paraxial rays make small angles, so Snell's law \(n_1\sin\theta_1=n_2\sin\theta_2\) reduces to \(n_1\theta_1=n_2\theta_2\); thin-prism deviation \(\delta=(\mu-1)A\) follows.
- Alternating Current — RMS values from \(\sin^2\omega t = \tfrac12(1-\cos 2\omega t)\) averaged over a cycle.
- Motion in a Plane — \(\sin 2\theta\) sets maximum projectile range and the equal-range complementary angles.
Frequently asked questions
The trigonometry questions NEET aspirants ask most often.
When can I use the small-angle approximation sin θ ≈ θ?
How do I convert degrees to radians and back?
What are the standard-angle values of sin, cos and tan?
Why must the angle be in radians for sin θ ≈ θ?
What is the sign of each trig ratio in the four quadrants?
How does the small-angle approximation make a pendulum simple harmonic?
What is the difference between sin(A+B) and sin A + sin B?
Trigonometry & approximations in one screen
- Radians: \(180^\circ=\pi\) rad; degrees→rad ×\(\pi/180\); use radians for calculus and all approximations; \(1\) rad ≈ \(57.3^\circ\).
- Ratios: \(\sin=\text{opp/hyp}\), \(\cos=\text{adj/hyp}\), \(\tan=\sin/\cos\); resolve as \(A_x=A\cos\theta,\ A_y=A\sin\theta\).
- Standard angles: sine \(=0,\tfrac12,\tfrac{1}{\sqrt2},\tfrac{\sqrt3}{2},1\); cosine the reverse; tangent undefined at \(90^\circ\).
- Quadrant signs (ASTC): Q-I all +, Q-II sin +, Q-III tan +, Q-IV cos +.
- Identities: \(\sin^2\theta+\cos^2\theta=1\); \(\sin(A\pm B)\), \(\cos(A\pm B)\) with cross terms; \(\sin 2\theta=2\sin\theta\cos\theta\); \(\cos 2\theta=1-2\sin^2\theta\).
- Small angles (θ in rad, θ ≪ 1): \(\sin\theta\approx\theta\), \(\tan\theta\approx\theta\), \(\cos\theta\approx1-\theta^2/2\) — the basis of the pendulum, paraxial optics and gentle banking.