Branch of mathematics

Trigonometry

Sine, cosine and tangent — from the right triangle to the unit circle: a height and a gradient from one angle, and then everything that repeats.

Topics in this branch

Intermediate
Sine, cosine and tangent
Sine, cosine and tangent are three ratios of the sides of a right triangle. See why they depend on the angle alone, where the exact values for 30°, 45° and 60° come from, and how to find a side you cannot measure.
Intermediate
The unit circle
The unit circle carries the sine and the cosine from a triangle onto the whole circle. See how the coordinates of a point become the values of the functions, what signs they take in the four quadrants, and what a negative angle or one past a full turn means.
Advanced
The Pythagorean identity
sin²α + cos²α = 1 is the Pythagorean theorem written on the unit circle. See where it comes from, how to find the cosine from the sine, and why an answer is incomplete without knowing the quadrant.
Advanced
The laws of sines and cosines
The law of sines and the law of cosines carry trigonometry into any triangle — including one with no right angle. See which to pick for the data you are given, why the law of cosines generalises Pythagoras, and when there are two solutions.
Advanced
Trigonometric graphs
A sine wave is the unit circle unrolled along an axis. See where the period of 2π comes from, why the cosine is a shifted sine, and what happens to the graph of the tangent where the function does not exist.
Advanced
Radian measure
A radian is the angle that cuts an arc equal to the radius. See where the 2π of a full turn comes from, how to convert degrees and radians both ways, and why the arc-length and sector-area formulas collapse to a single multiplication in this measure.
Advanced
Reduction formulas
Reduction formulas bring the sine, cosine and tangent of any angle back to an acute one. Instead of memorising a dozen identities, see the single rule behind them: at 180° and 360° the function stays, at 90° and 270° it swaps for its cofunction, and the sign comes from the quadrant.
Advanced
Angle sum and difference
The sine of a sum is not the sum of the sines. See where sin(α ± β) and cos(α ± β) come from, how the double-angle formulas fall out of them, how to get the exact value of sin 75°, and how to use them to prove trigonometric identities.
Advanced
Trigonometric equations
The equation sin x = 1/2 has infinitely many solutions, because the sine repeats. See how to find the base solution, how to write all the others with a parameter k, and how to pick out the ones that fall inside a given interval.
Advanced
Inverse trigonometric functions
Arcsine, arccosine and arctangent answer the reverse question: which angle has this value. See why the domain had to be restricted, what the ranges of the three functions are, and when arcsin(sin x) is not x.

Why this branch is worth learning

Every topic here settles a real situation. One example from each lesson:

  • Wheelchair ramps
    The step up to the terrace is 45 cm and a wheelchair needs a ramp no steeper than 6%. A gradient is the tangent of the angle, so tan α = 0.06, meaning α ≈ 3.4°, and the horizontal run comes to 45 cm / 0.06 = 750 cm. The ramp has to be 7.5 m long — usually more than the space in front of the terrace, which is why such ramps end up as two shorter flights with a landing.
    Sine, cosine and tangent
  • The workshop
    Six holes spaced evenly on a circle of radius 12 cm sit 60° apart. Rather than bending a tape along the arc, work out the coordinates: the third hole falls at 120°, that is at (12 · cos 120°, 12 · sin 120°) = (−6 cm, 10.4 cm). The minus sign says it lies 6 cm to the left of the centre, and measuring along two perpendicular directions lands it to the millimetre.
    The unit circle
  • Roofing
    Plans give the pitch of a roof as a percentage: 40% means tan α = 0.40. The identity 1 + tan²α = 1/cos²α turns that into cos α = 1/√1.16 = 0.93, so the slope above a 6 m span measures 6 / 0.93 = 6.46 m. Order sheeting off the plan width and every course falls 46 cm short — and no protractor came near the roof.
    The Pythagorean identity
  • Fencing a plot
    Two sides of a plot run 12 m and 9 m and meet at 115°. The law of cosines closes the corner: √(12² + 9² − 2 · 12 · 9 · cos 115°) = √(225 + 91.3) = 17.8 m. The cosine of an obtuse angle is negative, so the subtraction turns into an addition — 2.8 m more fencing than the 15 m a square corner would have needed.
    The laws of sines and cosines
  • Mains electricity
    Mains voltage does not sit at 230 V — it runs along a sine, u(t) = 325 · sin(2π · 50 · t). The 230 V figure is the peak divided by √2, so the wires really do carry 325 V at the top of each swing. At 50 Hz one period lasts 0.02 s and the voltage passes through zero 100 times a second, which is the 100 Hz flicker you catch out of the corner of your eye in cheap LED bulbs.
    Trigonometric graphs
  • The socket in the wall
    Mains voltage runs as a sine, u(t) = 325 · sin(ωt), where ω = 2π · 50 Hz ≈ 314 rad/s. Four milliseconds after a zero crossing the angle is 314 · 0.004 = 1.256 rad and the voltage is 325 · sin(1.256) ≈ 309 V. The calculator has to be in RAD mode for that: in DEG it computes sin 1.256° ≈ 0.022 and answers 7 V instead of 309 V.
    Radian measure
  • Noise-cancelling headphones
    A microphone picks up engine noise of amplitude 40 mPa and the speaker plays the same tone shifted by 180°: sin(x + 180°) = −sin x, so the reproduced wave is exactly opposite. The sum 40 · sin x + 40 · sin(x + 180°) = 0 — silence. Active noise cancellation is one reduction formula, executed 48,000 times a second.
    Reduction formulas
  • Projectile range
    The range of a projectile launched at speed v and angle α is R = v² · sin 2α / g — a formula that comes straight out of the double-angle identity. For v = 20 m/s and α = 30° that is 400 · sin 60° / 9.81 = 35.3 m, and for α = 45° it is 40.8 m. Since sin 2α peaks at 2α = 90°, the best launch angle is always 45° — which is why a shot put flies furthest at exactly that angle.
    Angle sum and difference
  • Tides
    The depth in a harbour varies as h(t) = 2.5 + 1.8 · sin(2πt/12.4) metres, with t in hours. A yacht drawing 3.4 m can enter when sin(2πt/12.4) ≥ 0.5, that is when 2πt/12.4 lies between π/6 and 5π/6 — a window from t = 1.03 h to t = 5.17 h, four hours and eight minutes of high water. Harbour authorities publish exactly this number as the "tidal window".
    Trigonometric equations
  • Road gradients
    A road sign reading "8%" means the road climbs 8 m over 100 m of horizontal distance. The angle is arctan 0.08 = 4.57°. A 12% climb is arctan 0.12 = 6.84°, and the maximum permitted gradient of a wheelchair ramp (6%) corresponds to arctan 0.06 = 3.43°. Without the arctangent a percentage cannot be turned into an angle at all.
    Inverse trigonometric functions

Branch formulas

Branch: Trigonometry

Sine, cosine and tangent

  • Sine

    sin⁡α=ac\sin \alpha = \frac{a}{c}

    the leg opposite the angle over the hypotenuse

  • Cosine

    cos⁡α=bc\cos \alpha = \frac{b}{c}

    the leg adjacent to the angle over the hypotenuse

  • Tangent

    tan⁡α=ab=sin⁡αcos⁡α\tan \alpha = \frac{a}{b} = \frac{\sin \alpha}{\cos \alpha}

    the opposite leg over the adjacent leg

  • The opposite leg

    a=c⋅sin⁡αa = c \cdot \sin \alpha

    the definition of the sine solved for a

  • The adjacent leg

    b=c⋅cos⁡αb = c \cdot \cos \alpha

    the definition of the cosine solved for b

  • The hypotenuse

    c=asin⁡αc = \frac{a}{\sin \alpha}

    when the leg opposite the angle is known

  • Special angles

    sin⁡30∘=12,sin⁡45∘=22,sin⁡60∘=32\sin 30^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}

    exact values, not readings from a table

The unit circle

  • Equation of the unit circle

    x2+y2=1x^2 + y^2 = 1

    a circle centred at (0, 0) with radius 1

  • The point on the circle

    P=(cos⁡α,  sin⁡α)P = (\cos \alpha, \; \sin \alpha)

    the cosine is the x-coordinate, the sine the y-coordinate

  • Coterminal angles

    sin⁡(α+360∘)=sin⁡α\sin(\alpha + 360^\circ) = \sin \alpha

    a full turn changes nothing

The Pythagorean identity

  • The Pythagorean identity

    sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1

    holds for every angle, with no exceptions

  • Sine from cosine

    sin⁡α=±1−cos⁡2α\sin \alpha = \pm\sqrt{1 - \cos^2 \alpha}

    the quadrant picks the sign

  • Cosine from sine

    cos⁡α=±1−sin⁡2α\cos \alpha = \pm\sqrt{1 - \sin^2 \alpha}

    the same identity solved for the cosine

  • Tangent as a quotient

    tan⁡α=sin⁡αcos⁡α\tan \alpha = \frac{\sin \alpha}{\cos \alpha}

    requires cos α ≠ 0

  • A derived identity

    1+tan⁡2α=1cos⁡2α1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha}

    the identity divided through by cos²α

The laws of sines and cosines

  • The law of sines

    asin⁡α=bsin⁡β=csin⁡γ\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}

    a side and the angle opposite it — always as a pair

  • A side from the law of sines

    b=a⋅sin⁡βsin⁡αb = \frac{a \cdot \sin \beta}{\sin \alpha}

    the proportion solved for the unknown side

  • The law of cosines

    a2=b2+c2−2bccos⁡αa^2 = b^2 + c^2 - 2bc \cos \alpha

    α is the angle between the sides b and c

  • An angle from three sides

    cos⁡α=b2+c2−a22bc\cos \alpha = \frac{b^2 + c^2 - a^2}{2bc}

    the law of cosines solved for the angle

  • The angle sum

    α+β+γ=180∘\alpha + \beta + \gamma = 180^\circ

    the third angle always follows from the other two

  • Area of a triangle

    P=12absin⁡γP = \frac{1}{2} ab \sin \gamma

    two sides and the angle BETWEEN them — no height needed

Trigonometric graphs

  • The sine wave

    y=sin⁡xy = \sin x

    the y-coordinate of the point on the unit circle, as a function of the angle

  • The cosine wave

    y=cos⁡xy = \cos x

    the x-coordinate of the same point

  • Periodicity

    sin⁡(x+2π)=sin⁡x\sin(x + 2\pi) = \sin x

    the sine and the cosine have period 2π

  • The cosine as a shifted sine

    cos⁡x=sin⁡(x+π2)\cos x = \sin\left(x + \frac{\pi}{2}\right)

    the same shape, moved by a quarter period

  • The period of the tangent

    tan⁡(x+π)=tan⁡x\tan(x + \pi) = \tan x

    π, half as long as the sine’s

  • Asymptotes of the tangent

    x=π2+kπx = \frac{\pi}{2} + k\pi

    there the cosine vanishes and the function does not exist

Radian measure

  • One radian

    1 rad=180∘π≈57.30∘1\ \text{rad} = \frac{180^\circ}{\pi} \approx 57.30^\circ

    the angle whose arc is as long as the radius

  • A full turn and a half turn

    360∘=2π rad,180∘=π rad360^\circ = 2\pi\ \text{rad}, \qquad 180^\circ = \pi\ \text{rad}

    a circumference of 2πr holds 2π radii

  • Degrees → radians

    αrad=αdeg⁡⋅π180∘\alpha_{\text{rad}} = \alpha_{\deg} \cdot \frac{\pi}{180^\circ}

    the result is a smaller number than the degree count

  • Radians → degrees

    αdeg⁡=αrad⋅180∘π\alpha_{\deg} = \alpha_{\text{rad}} \cdot \frac{180^\circ}{\pi}

    π cancels and the degree count is left

  • Arc length

    l=rαl = r \alpha

    α in radians — no conversion factor at all

  • Sector area

    P=12r2αP = \frac{1}{2} r^2 \alpha

    α in radians; α = 2π gives πr²

Reduction formulas

  • Supplement of the angle

    sin⁡(180∘−α)=sin⁡α,cos⁡(180∘−α)=−cos⁡α\sin(180^\circ - \alpha) = \sin \alpha, \qquad \cos(180^\circ - \alpha) = -\cos \alpha

    quadrant II: sine positive, cosine negative

  • Half a turn onwards

    sin⁡(180∘+α)=−sin⁡α,cos⁡(180∘+α)=−cos⁡α\sin(180^\circ + \alpha) = -\sin \alpha, \qquad \cos(180^\circ + \alpha) = -\cos \alpha

    quadrant III: both functions negative

  • Completing a full turn

    sin⁡(360∘−α)=−sin⁡α,cos⁡(360∘−α)=cos⁡α\sin(360^\circ - \alpha) = -\sin \alpha, \qquad \cos(360^\circ - \alpha) = \cos \alpha

    quadrant IV: sine negative, cosine positive

  • A negative angle

    sin⁡(−α)=−sin⁡α,cos⁡(−α)=cos⁡α\sin(-\alpha) = -\sin \alpha, \qquad \cos(-\alpha) = \cos \alpha

    the sine is odd, the cosine even

  • Complement of the angle

    sin⁡(90∘−α)=cos⁡α,cos⁡(90∘−α)=sin⁡α\sin(90^\circ - \alpha) = \cos \alpha, \qquad \cos(90^\circ - \alpha) = \sin \alpha

    the function swaps for its cofunction

  • A quarter turn onwards

    sin⁡(90∘+α)=cos⁡α,cos⁡(90∘+α)=−sin⁡α\sin(90^\circ + \alpha) = \cos \alpha, \qquad \cos(90^\circ + \alpha) = -\sin \alpha

    the functions swap again, sign from quadrant II

  • Period of the tangent

    tan⁡(180∘+α)=tan⁡α\tan(180^\circ + \alpha) = \tan \alpha

    the tangent repeats every half turn

Angle sum and difference

  • Sine of a sum

    sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta

    the sign on the right matches the one on the left

  • Sine of a difference

    sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta

    the same formula with β replaced by −β

  • Cosine of a sum

    cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta

    careful: the sign on the right is REVERSED

  • Cosine of a difference

    cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta

    the starting formula — the other three follow from it

  • Tangent of a sum

    tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}

    the quotient of the two formulas above

  • Sine of a doubled angle

    sin⁡2α=2sin⁡αcos⁡α\sin 2\alpha = 2 \sin \alpha \cos \alpha

    the sum formula with β = α

  • Cosine of a doubled angle

    cos⁡2α=cos⁡2α−sin⁡2α=1−2sin⁡2α=2cos⁡2α−1\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha = 1 - 2\sin^2 \alpha = 2\cos^2 \alpha - 1

    three forms of one value — pick the one that fits the data

Trigonometric equations

  • Equation with a sine

    sin⁡x=c  ⟺  x=α+2kπ ∨ x=π−α+2kπ\sin x = c \iff x = \alpha + 2k\pi \ \lor \ x = \pi - \alpha + 2k\pi

    α is the base solution; k is any integer

  • Equation with a cosine

    cos⁡x=c  ⟺  x=α+2kπ ∨ x=−α+2kπ\cos x = c \iff x = \alpha + 2k\pi \ \lor \ x = -\alpha + 2k\pi

    the solutions sit symmetrically about the x axis

  • Equation with a tangent

    tan⁡x=c  ⟺  x=α+kπ\tan x = c \iff x = \alpha + k\pi

    one family only — the tangent has period π, not 2π

  • Condition for a solution

    ∣c∣>1  ⟹  sin⁡x=c has no solution|c| > 1 \implies \sin x = c \ \text{has no solution}

    the same for the cosine; the tangent takes every value

  • Special cases

    sin⁡x=0  ⟺  x=kπ\sin x = 0 \iff x = k\pi

    and cos x = 0 ⟺ x = π/2 + kπ

  • Reducing to one function

    sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2\sin x \cos x

    the doubled angle broken up before factoring

Inverse trigonometric functions

  • Arcsine

    y=arcsin⁡x  ⟺  sin⁡y=x  and  y∈[−π2,π2]y = \arcsin x \iff \sin y = x \ \text{ and } \ y \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]

    domain [−1, 1], range [−π/2, π/2]

  • Arccosine

    y=arccos⁡x  ⟺  cos⁡y=x  and  y∈[0,π]y = \arccos x \iff \cos y = x \ \text{ and } \ y \in \left[0, \pi\right]

    domain [−1, 1], range [0, π]

  • Arctangent

    y=arctan⁡x  ⟺  tan⁡y=x  and  y∈(−π2,π2)y = \arctan x \iff \tan y = x \ \text{ and } \ y \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)

    domain: every real number

  • Composition with the arcus inside

    sin⁡(arcsin⁡x)=xfor x∈[−1,1]\sin(\arcsin x) = x \quad \text{for } x \in [-1, 1]

    always holds within the domain of the arcus

  • Composition with the arcus outside

    arcsin⁡(sin⁡x)=xonly for x∈[−π2,π2]\arcsin(\sin x) = x \quad \text{only for } x \in \left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]

    outside that interval the result is a DIFFERENT angle

  • Arcsine and arccosine together

    arcsin⁡x+arccos⁡x=π2\arcsin x + \arccos x = \frac{\pi}{2}

    straight from the reduction formula for the complement

Related articles

A luminous unit circle with a radius sweeping out an angle, a sine wave unrolling from it as a streak of light, beside a right triangle drawn in stars, against a cosmic nebula in violet and magenta
Trigonometry

Why sine doesn't depend on the size of the triangle

The height of a tower, the width of a river, the distance to a ship offshore — all of it can be computed without going anywhere near the thing being measured. The whole of trigonometry rests on one observation: the ratio of two sides in a right triangle depends only on the angle, not on how big the triangle is. That is what turns an angle into a number — and lets it be tabulated once and for all.

13 min