Branch of mathematics

Trigonometry

Sine, cosine and tangent — from the right triangle through the unit circle to the graphs of periodic functions.

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 why the radian is the natural measure of an angle.
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.

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

  • Degrees → radians

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

    180° is π radians

  • Radians → degrees

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

    the same relation the other way round

  • 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

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

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