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Right-triangle trigonometry (SOH-CAH-TOA)

Sine, cosine and tangent as side ratios, and solving right triangles.

In a right triangle, sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Knowing one angle and one side fixes everything. Picture it: the 5-12-13 triangle below is drawn to scale; the ratios are its sides. Think it: because all right triangles with a given angle are similar, the ratio depends only on the angle — that is why sin θ is a function.

Urugero: hypotenuse of 5 and 12

Geometry · Pythagorean-Theorem

Hypotenuse of 5 and 12

5,\ 12

ku

  1. c^2 = a^2 + b^2 = 5^2 + 12^2 = 169

    Pythagoras: in a right triangle the square on the hypotenuse equals the sum of the squares on the legs.

  2. c = \sqrt{169} = 13

    Take the square root.

i
c = 13

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
\theta
theta
The usual name for an angle.
^\circ
degrees
1/360 of a full turn. 180° = π radians.
\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
\arcsin,\ \sin^{-1}
inverse sine
The angle whose sine is the given value (and likewise arccos, arctan).

How to: Right-triangle trigonometry (SOH-CAH-TOA)

  1. Label the sides relative to the angle you know: opposite, adjacent, hypotenuse.
  2. Pick the ratio that links the side you know to the side you want.
  3. Solve the resulting equation; use the inverse function when the unknown is the angle.

Questions people ask

Why radians instead of degrees?

A radian is the angle whose arc equals the radius, so it is a pure ratio rather than an arbitrary 1/360 of a turn. With radians the derivative of sin x is exactly cos x; with degrees a factor of π/180 appears everywhere.

Why does sin x = 1/2 have infinitely many solutions?

Sine repeats every full turn, and within one turn it reaches 1/2 twice (at π/6 and 5π/6). Add any whole number of turns to either and it is still true.

How do I remember the exact values?

Two triangles: the 45-45-90 with sides 1, 1, √2 and the 30-60-90 with sides 1, √3, 2. Every value for 30°, 45° and 60° is a ratio of those sides; symmetry gives the rest of the circle.

Kuri Gukoresha

in Trigonometry

The unit circleTrigonometric equationsTrigonometric identitiesDegrees and radiansLaw of sines and law of cosinesGraphs of sine, cosine and tangentInverse trigonometric functions