Mipango

Kikokotoo cha Utambulisho wa Trigonometria: Thibitisha Utambulisho

Kikokotoo hiki kinakagua kama mlinganyo kama sin²x + cos²x = 1 ni utambulisho wa kweli. Kinakokotoa pande zote mbili kwa pembe nyingi; thamani yoyote ikitofautiana kinaonyesha mfano pinzani, na isipotofautiana kinathibitisha utambulisho — kisha fomula zilizo hapa chini zinakusaidia kuandika uthibitisho.

Tumia x kwa pembe; sin, cos, tan, sec, csc na cot zote zinafanya kazi, na sin^2(x) maana yake ni (sin x)².

Tokeo

Weka thamani kisha bonyeza Kokotoa ili kuona tokeo.

Fomula

sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1

The Pythagorean identity; dividing by cos²x gives 1 + tan²x = sec²x.

tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}

The quotient identity, for cos x ≠ 0.

sin⁡2x=2sin⁡xcos⁡x,cos⁡2x=cos⁡2x−sin⁡2x\sin 2x = 2\sin x\cos x, \quad \cos 2x = \cos^2 x - \sin^2 x

Double-angle identities.

Jinsi ya kutumia kikokotoo hiki

  1. Type the identity with an equals sign, using x for the angle — for example tan(x)*cos(x) = sin(x).
  2. Press Calculate. Both sides are evaluated at 48 angles between about −6 and 6 radians, skipping any where a side is undefined.
  3. If the sides ever differ, you get the angle where they differ — the identity is false. If they always agree, it is an identity, and you can prove it with the formulas below.

Mifano iliyotatuliwa

The Pythagorean identity

Both sides are evaluated at 48 angles; they agree every time.

Proof: on the unit circle a point is (cos x, sin x), and by Pythagoras cos²x + sin²x = 1.

Verify a quotient identity

tan x · cos x = (sin x / cos x) · cos x = sin x, wherever cos x ≠ 0.

The calculator confirms it at every angle it tested.

Using sec and tan

Divide sin²x + cos²x = 1 by cos²x: tan²x + 1 = sec²x, so sec²x − tan²x = 1.

A false “identity”

At x = −6.1: sin(−12.2) ≈ 0.3582 but 2 sin(−6.1) ≈ 0.3643, so the two sides differ.

The correct double-angle identity is sin 2x = 2 sin x cos x.

Makosa ya kawaida

  • Treating sin 2x as 2 sin x — the 2 is inside the function.
  • Writing sin x² when you mean (sin x)²; type sin(x)^2 or sin^2(x).
  • Working on both sides at once in a proof: transform one side until it matches the other.
  • Dividing by an expression such as cos x without noting where it is zero.

Maswali yanayoulizwa mara kwa mara

How does the trig identities calculator verify an identity?

It evaluates both sides at 48 different angles. One mismatch proves the equation is not an identity; agreement everywhere is strong evidence that it is — and then the formulas help you write a proof.

Is a numerical check a proof?

No. It can prove an identity false (with a counterexample) but only confirm a true one. Teachers expect an algebraic proof using identities such as sin²x + cos²x = 1.

Which trig identities should I learn first?

The Pythagorean identity sin²x + cos²x = 1, tan x = sin x / cos x, and the double-angle formulas for sin 2x and cos 2x.

Does it work in degrees?

The angle is in radians, but an identity is true in both units or in neither, so the verdict is the same.