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Trigonometric identities
Pythagorean, double-angle and sum identities, verified by simplification.
Identities are equations true for every angle. The Pythagorean identity sin²x + cos²x = 1 is Pythagoras on the unit circle; the double-angle and sum formulas follow from it and rotation. To verify one, simplify the difference of the two sides to zero.
Töötatud näide · simplify sin(x)^2 + cos(x)^2
Simplify sin(x)^2 + cos(x)^2
Samm-sammult
- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}
Start from the expression.
- 1
Simplify.
Vastuse avalikustamine
Nüüd sina. Vali probleem, või kirjuta või joonista oma. Iga samm, pilt, vastus peidetud, kuni küsid.
Tasuta konto lisab märkmeid igale õppetunnile, ülestähenduse, oma lahendatud probleemide kohta ühes kohas ja juhendaja kohta, kellelt võid küsida selle lehekülje kohta. Matemaatika ise on avatud kõigile, allkirjastatud või mitte.
Registreeru SisselogimineSiin kasutatavad sümbolid
Koputage kõiki sümboleid, mis on mõeldud täieliku definitsiooni, pildi ja iga tähe all.
Kuidas: Trigonometric identities
- Start from the expression.
- Simplify.
Küsimusi, mida inimesed küsivad
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.
Rohkem Trigonometry
The unit circleTrigonometric equationsDegrees and radiansRight-triangle trigonometry (SOH-CAH-TOA)Law of sines and law of cosinesGraphs of sine, cosine and tangentInverse trigonometric functions