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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.

Vinna dæmi · hypotenuse of 5 and 12

Geometry · Pythagorean-Theorem

Hypotenuse of 5 and 12

5,\ 12

Skref fyrir skref

  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.

Sýna svarið
c = 13

Núna ūú Veldu vandamál, eða sláðu inn eða teikna eigin. Hvert skref, mynd, svarið falið þar til þú spyrð.

Haltu eigin vinnu þinni

Ókeypis reikningur bætir við athugasemdum við hverja kennslustund, skrá yfir það sem þú hefur lokið, leyst vandamál á einum stað og kennari sem þú getur spurt um þessa síðu.Stærðfræðin sjálf er opin öllum, skráður inn eða ekki.

Skráðu þig inn Innskráning

Tákn notuð hér

Pikkaðu á hvaða tákn sem er fyrir fulla skilgreiningu, mynd og hvað hvert bréf í því merkir.

Hvernig á að nota: 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.

Spurningar sem fólk spyr

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.

Meira í Trigonometry

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