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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.

Nhazi ahụ ejirila · simplify sin(x)^2 + cos(x)^2

Algebra · Simplifying

Simplify sin(x)^2 + cos(x)^2

\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}

Nzọụkwụ site n'ụdị

  1. \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}

    Start from the expression.

  2. 1

    Simplify.

Gosi nzaghachi
1

Ugbu a ị Họrọ nsogbu, mọọbụ tinye mọọbụ depụta gị. Nzọụkwụ ọbụla, inyogo, azịza ahụ echekwara ruo mgbe ịjụla.

Kwụsị ọrụ gị

Akwụkwọ ndebanye n'efu na-egbakwunye ntọala n'ụdị ọmụmụ ihe ọbụla, ndepụta nke ihe ị ga-eme, nsogbu ị ga-ewepụ n'otu ebe, na onye nkụzi ị ga-ajụ banyere ihuakwụkwọ a. Akwụkwọ ndepụta ihe bụ nke ọbụla mepere maka onye ọbụla, ọbụla ma ọ bụ na ọ bụghị.

Akaụntụ Nbanye

Aịkọn ndị ahụ ejirila ebe a

Tinye ọbụla simbụl maka nkọwa zuru ezu, inyogo, na ihe ọbụla ọbụla na-egosi.

Olee otú: Trigonometric identities

  1. Start from the expression.
  2. Simplify.

Ajụjụ ndị mmadụ jụrụ

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.

Oge 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