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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.

Umzekelo osebenzelayo · hypotenuse of 5 and 12

Geometry · Pythagorean-Theorem

Hypotenuse of 5 and 12

5,\ 12

Inyathelo ngenyathelo

  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.

Bonisa impendulo
c = 13

ngoku Khetha ingxaki, okanye ubhale okanye uzobe yakho. Inyathelo ngalinye, umfanekiso, impendulo ifihlakele de ubuze.

Ukugcina umsebenzi wakho

I akhawunti ekhululekileyo idibanisa amaphetshana kwi ncwadi nganye, irekhodi lento ogqibe ngayo, iingxaki zakho ezisoliweyo kwindawo enye, nomfundi onokuthi ubuze malunga nale phepha. IiMathematiki ngokwazo zivuliwe kubo bonke, bangeniswe okanye hayi.

Bhalisa Igama elithile

Iimpawu ezisetyenziswa apha

Nqakraza nasiphi na isibonakaliso sokuqonda okupheleleyo, umfanekiso, nokuba iileta zonke zithetha ntoni.

Indlela yokusebenza: 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.

Imibuzo abantu bebuza

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.

IiNkqubo Trigonometry

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