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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.
ตัวอย่างที่ทำงานแล้ว · hypotenuse of 5 and 12
ก้าวต่อก้าว
- 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.
- c = \sqrt{169} = 13
Take the square root.
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วิธีการ: Right-triangle trigonometry (SOH-CAH-TOA)
- Label the sides relative to the angle you know: opposite, adjacent, hypotenuse.
- Pick the ratio that links the side you know to the side you want.
- Solve the resulting equation; use the inverse function when the unknown is the angle.
คำถามที่คนถาม
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.
เพิ่มเติมใน Trigonometry
The unit circleTrigonometric equationsTrigonometric identitiesDegrees and radiansLaw of sines and law of cosinesGraphs of sine, cosine and tangentInverse trigonometric functions