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The unit circle
Exact values of sine and cosine at the special angles.
On a circle of radius 1, the coordinates of the point at angle θ are (cos θ, sin θ). The angles 30°, 45°, 60° and their reflections give exact values built from √2 and √3 — worth knowing by heart, and all derivable from two triangles.
İşlediği örnek: sin(pi/3)
Adım adım
- \sin{\left(\frac{\pi}{3} \right)} = \frac{\sqrt{3}}{2}
Evaluate.
Cevabı açıkla.
Symbols used here
The non-negative number whose square (n-th power) is x.
Ratio of a circle's circumference to its diameter, 3.14159…
Ratios of sides in a right triangle; coordinates on the unit circle.
Equal to the precision shown, not exactly.
Inequalities that allow equality; < and > exclude it.
The usual name for an angle.
1/360 of a full turn. 180° = π radians.
The angle whose sine is the given value (and likewise arccos, arctan).
How to: The unit circle
- Evaluate.
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.
Kendini dene.
Daha fazlası Trigonometry
Trigonometric equationsTrigonometric identitiesDegrees and radiansRight-triangle trigonometry (SOH-CAH-TOA)Law of sines and law of cosinesGraphs of sine, cosine and tangentInverse trigonometric functions